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How to Write the Cube Root in XYZ Homework

As a student who uses the XYZ Homework, you must know that a few types of questions require a mathematical expression or equation in the answer box. Due to the fact that XYZ Homework follows order of operations, the use of proper grouping symbols is important.

One of these symbols is the cube root. As a student, you must know how to write the cube root in XYZ Homework. Even if it is a must, it is normal if you have no idea how to write it. So, how to write the cube root in XYZ Homework?

For those who want to know the way to write the cube root in XYZ Homework, you will have to type 221b and press alt and x keys.

Aside from the way to write the cube root in XYZ Homework, you might also want to know how to write the others. Here is the information for you to read:

Operation symbols:

• Meaning: Addition Symbol: + Notation: x + 3 Display: x + 3
• Meaning: Subtraction Symbol: – Notation: x-3, -5 Display: x – 3, -5
• Meaning: Multiplication Symbol: * Notation: 3*10, 3x Display: 3*10, 3x
• Meaning: Division Symbol: / Notation: x/3 Display: x/3
• Meaning: Exponents Symbol: ^ Notation: x^3 Display: x3

Relation symbols:

• Meaning: Equal to Symbol: = Notation: y=5 Display: y = 5
• Meaning: Not equal to Symbol: ! = Notation: y!=-3 Display: y≠ -3
• Meaning: Greater than Symbol: > Notation: x>3 Display: x > 3
• Meaning: Less than Symbol: < Notation: x<4 Display: x <4
• Meaning: Greater than or equal to Symbol: >= Notation: x>=-3 Display: x ≥ -3
• Meaning: Less than or equal to Symbol: <= Notation: y<=-10 Display: y ≤ -10

Grouping symbols:

• Meaning: Parentheses Symbol: () Notation: 3/ (x+5) Display: 3/x + 5
• Meaning: Nested parentheses Symbol: (( )) Notation: 3/(2(x+5)) Display: 3/2(x + 5)
• Meaning: Brackets Symbol: [] Notation: [5, 9] Display: [5, 9]
• Meaning: Braces Symbol: {} Notation: {1, 2, 3, 4} Display: {1, 2, 3, 4}

Function symbols:

• Meaning: Square root Symbol: sqrt() Notation: sqrt(7) Display: √7
• Meaning: nth root Symbol: root(n)() Notation: root(3)(27x) Display: 3 √ — 27x
• Meaning: Absolute value Symbol: abs() Notation: abs(−5) Display:  –5 
• Meaning: Factorial Symbol: ! Notation: 5! Display: 5!
• Meaning: Base 10 logarithm Symbol: log() Notation: log(5) Display: log 5
• Meaning: Natural log Symbol: In () Notation: In (x) Display: In x
• Meaning: General base a Symbol: log_a () Notation: log_8 (x) Display: log8x

Miscellaneous symbols:

• Meaning: π Symbol: pi Notation: pi r^2 Display: πr 2
• Meaning: e Symbol: e Notation: e^x Display: ex
• Meaning: Infinity Symbol: oo Notation: (−oo,0) Display: (–∞, 0)
• Meaning: Does Not Exist Symbol: DNE Notation: DNE Display: DNE

Advanced examples:

• Topic: Enter a reduced fraction Example: 3/5 Display: 3/5
• Topic: Enter a mixed number Example: 2_3/5 Display:  2 3/5
• Topic: Enter a linear equation Example: y=3x+5 Display: y = 3x + 5
• Topic: Rational expression 1 Example: 2/(3x+5) Display: 2/3x +5
• Topic: Rational expression 2 Example: 2/3x+5 Display: 2/3x +5
• Topic: Rational expression 3 Example:  2/ ((x+5)(x–3)) Display: 2/ ((x+5)(x–3))
• Topic: Exponential expression 1 Example: 3^x+2 Display:  3x + 2
• Topic: Exponential expression 2 Example: 3^(x+2) Display:  3x + 2

For every student who uses an easy to use online teaching tool to study online called XYZ Homework, it is a must for you to know how to use XYZ Homework. These followings will uncover about entering answers and the preview button. Please read everything well and if you need, you can also write down the important information on a piece of paper.

Entering answers:

It is important for you to input your answers correctly when completing assignments in XYZ Homework. The reason why it is important is because it is useful to help the system determine their accuracy. There are a total of two ways to enter answers. The first one is by calculator style math or ASCII math reference on the inside covers and the second one is by using the MathQuill equation editor.

MathQuill is the one that allows the students to enter answers as correctly as displayed mathematics. Generally, when you click in the answer box following every question, there will be a yellow box with an arrow showing up on the right side of the box. By clicking this arrow button, you will be allowed to open a pop up window with the MathQuill tools for inputting normal math symbols.

Preview button:

Beside the answer box, usually, there will be a preview button. Clicking this button makes the system display how it interprets the answer keyed in. It should be noted that the preview button is not grading your answer. This one is just indicating if the format is correct.

Rationalize the denominator in the following:

√a – 1/√a  + 1 = (a-2 sqrt(a)+1)/(a-1)

Preview: a – 2√a + 1/a – 1 x syntax ok

Tips are the ones that will indicate the kind of answer the system is expecting. Please pay extra attention to these tips when entering fractions, mixed numbers, and ordered pairs.

For more information, please visit the official website of XYZ Homework. If you do not know its official website, it is  xyzhomework.com . Feel free to visit the official website whenever you want. If you have some questions to ask, please direct your questions and comments to  [email protected] . For those who are registered users, you can visit the support page for additional information.

XYZ Homework is the name of an easy to use online teaching pool. This one is full of features. It helps every student who wants to learn online. This tool is the one that provides powerful online instructional tools for faculty and students. Its unified learning environment mixes online assessment with MathTV.com video lessons and McKeague’s proven developmental math textbooks. All of them team up to reinforce the concepts that are taught in the classroom. There are some randomized questions that provide unlimited practice and instant feedback with all the benefits of automatic grading.

On my daily job, I am a software engineer, programmer & computer technician. My passion is assembling PC hardware, studying Operating System and all things related to computers technology. I also love to make short films for YouTube as a producer. More at about me…

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Cubes and Cube Roots

• SchoolTutoring Academy
• November 5, 2012

The cube of a number a is denoted by a 3 and it is nothing but the product of a for 3 rimes.

So, a 3 = a x a x a

2 3 = 2x2x2 = 8

5 3 = 5x5x5 = 125

(1)    If a 3 =b then the unit digit of all numbers ending with a is b.

(2)    The sum of n natural odd numbers is n 3 .

(3)    If a given number is a perfect cube , then its prime factors will always occur in groups of three.

Cube root is the other way of cube. a is called the cube root of b if a 3 =b.

i.e. if 3 3 =27 then 3 is called cube root of 27.

The cube root of a number can be found using estimating or prime factorization method.

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Algebra: Square Roots

Operations on polar form of a complex number.

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Here is everything you need to know about cube roots. You will learn what a cube root is, what a cube number is, and how to simplify expressions using cube roots.

Students first work with cube roots in 8 th grade, when they work with square roots and expand this knowledge as they progress through high school.

What is a cube number and a cube root?

A cube number is a number or variable that is ‘cubed’ which means it is multiplied by itself three times.

The cube root of a number is the value being multiplied by itself three times. Cube rooting a number is the inverse operation of cubing a number.

The cube root function looks like this \sqrt[3]{\quad} where 3 is the index of the root. The cube root sign can also be called a radical sign.

The cube root can also be expressed with the exponent, \cfrac{1}{3} \, .

For example, 4 \times 4 \times 4=64

This equation can be rewritten to be 4^3=64.

4 is the cube root, and 64 is the perfect cube number. This is because 4 is the number being cubed or multiplied by itself three times, and 64 is the product, so it is the perfect cube.

Therefore, the cube root of 64 is 4 which can be written as

\sqrt[3]{64}=4 or (64)^{\frac{1}{3}}=4

The table below has some of the perfect cube numbers and cube roots.

So, anything multiplied by itself three times forms a perfect cube.

x^3 is a perfect cubed algebraic expression where x is the cube root.

It can be useful to see how a cube number relates to an actual cube.

Let’s look at:

Each cubed relationship can be represented as an array which forms the shape of a cube that has a length 2 units, width 2 units, and depth 2 units, etc.

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Cubing negative numbers

Any number can be cubed including decimals, fractions and integers.

For example:

\cfrac{1}{3} \, \times \cfrac{1}{3} \, \times \cfrac{1}{3} \, =\cfrac{1}{27} \, \rightarrow \cfrac{1}{27} \, is a perfect cube.

(-5) \times(-5) \times(-5)=(-125) \rightarrow-125 is a perfect cube.

You will notice that when you cube a negative number, you get a negative number, and when you cube a positive number, you get a positive number.

This is because a negative number multiplied by a negative number multiplied by a negative number yields a negative result and a positive number multiplied by a positive number, multiplied by a positive number yields a positive result.

Likewise, the cube root of a positive number is a positive root and the cube root of a negative number is a negative root.

For example, \sqrt[3]{-125}=-5 (radical form) OR (-125)^{\frac{1}{3}}=-5 (exponential form)

Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8 Expressions and Equations (8.EE.A.1) Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 \times 3-5 = 3-3 = \cfrac{1}{33} = \cfrac{1}{27}
• Grade 8 Expressions and Equations (8.EE.A.1) Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational.
• High School Number and Quantity: The Real Number System (HSN-RN.A.1) Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define \cfrac{51}{3} to be the cube root of 5 because we want (\cfrac{51}{3})^3 = 5(\cfrac{1}{3})^3 \, to hold, so (\cfrac{51}{3})^3 must equal 5.

How to cube expressions

In order to cube expressions:

Take the expression and multiply it by itself three times.

Write the cubed expression.

Cube root examples

Example 1: cube positive number.

Cube the number 12.

2 Write the cubed expression.

Example 2: cube a negative fraction

Cube the number \left(-\cfrac{2}{3}\right).

Example 3: cube an expression

Cube the expression 2x^2.

You need to multiply 2 by itself three times and multiply x^2 by itself three times.

\begin{aligned} & 2 \times 2 \times 2=8 \\\\ & x^2 \times x^2 \times x^2=x^6 \end{aligned}

\left(2 x^2\right) \times\left(2 x^2\right) \times\left(2 x^2\right)=8 x^6

How to simplify cube roots

In order to simplify cube root expressions:

Look to see if the number or variable is a perfect cube.

If they are not perfect cubes, rewrite them with perfect cube factors.

Take the cube root of the perfect cubes.

Write the simplified answer.

Example 4: cube root of a number

Simplify \sqrt[3]{-512}.

512 is a perfect cube number because -8 \times -8 \times -8 = -512.

The number is a perfect cube number.

-8 is the simplified answer.

Example 5: cube root of non perfect cube number

Simplify the expression \sqrt[3]{54}.

54 is not a perfect cube.

54 does have a perfect cube factor. 54 can be rewritten as 27 \times 2, where 27 is a perfect cube.

\begin{aligned} & \sqrt[3]{27 \times 2}=\sqrt[3]{27} \times \sqrt[3]{2} \\\\ & \sqrt[3]{27}=3 \end{aligned}

\sqrt[3]{27}=3

\sqrt[3]{2} cannot take this cube root without a calculator because 2 is not a perfect cube.

The simplified answer is:

3 \sqrt[3]{2} (radical form) which also can be written as 3(2)^{\frac{1}{3}} (exponential form).

Example 6: cube root of an expression that is not a perfect cube

Simplify the expression \sqrt[3]{8 x^7}.

8 is a perfect cube and x^7 can be rewritten to be x^6 \times x^1. If the exponent is divisible by 3, it is considered to be a perfect cube.

If the exponent is not divisible by 3, rewrite the exponential expression so that one of the exponents is divisible by 3.

\sqrt[3]{x^1} \rightarrow this is not a perfect cube and can stay as \sqrt[3]{x^1} or can be written as x^{\frac{1}{3}}

\sqrt[3]{8 x^7}=2 x^2 \sqrt[3]{x} OR 2 x^2 \times x^{\frac{1}{3}} which can be simplified using laws of exponents to be 2 x^{\frac{7}{3}}

Teaching tips for cube numbers and cube roots

• When simplifying cube root expressions, students recall laws of exponents to help them to simplify the answer.
• Build upon prior knowledge of perfect square numbers and square roots to introduce cube numbers and cube roots.
• Although there are cube root calculators that students can access from their computers, have students simplify cube roots of given numbers or given expressions without a calculator to build number sense.
• Practice worksheets have their place in the classroom, however, infuse game playing or scavenger hunts for students to practice problems.

Easy mistakes to make

• Incorrect understanding of cube numbers For example, thinking that 2^3 is 2 \times 3=6 instead of 2 \times 2 \times 2=8
• Thinking that you can not take the cube root of a negative number For example, taking the square root of a negative number cannot be done in the real number system. However, taking the cube root of a negative number can be done in the real number system. \sqrt{-4}=\pm 2i (imaginary number) \sqrt[3]{-8}=-2

Related law of exponents lessons

• Law of exponents
• Dividing exponents
• Exponential notation
• Negative exponents
• Square root
• Anything to the power of 0
• Distributing exponents
• Exponent rules

Cube numbers and cube roots problems

1. Which number is a perfect cube?

64 is a perfect cube number because 4^3=4 \times 4 \times 4=64

2. The length of one side of a cube is 5 \; mm. What is the volume of the cube?

A cube has equal dimensions so if one side is 5 \mathrm{~mm} all sides are 5 \mathrm{~mm}.

To find the the volume of a cube, you need to multiply the \text{length} \times \text{width} \times \text{height} which in this case is 5 \times 5 \times 5=125.

So the volume of the cube is 125 \mathrm{~mm}^3.

3. Cube the expression -3 x^2.

-3 x^2 cubed is -3 x^2 \times-3 x^2 \times-3 x^2, \; -3 multiplied by itself three times is -27 and -x^2 multiplied by itself three times is -x^8.

You can apply laws of exponents when multiplying x^2 by itself three times.

4. What is \sqrt[3]{-729} \, ?

\sqrt[3]{-727} means to find the number that is multiplied to itself three times to get -729.

In this case (-9)^3=-9 \times-9 \times-9=-729

5. What is \sqrt[3]{27 x^9} \, ?

\sqrt[3]{27 x^9} can be rewritten as \left(27 x^9\right)^{\frac{1}{3}}. \; 27 is a perfect cube number because

3 \times 3 \times 3 = 27 and x^9 is also a perfect cube expression because the exponent is divisible by 3.

You can also apply law of exponents to simplify \left(x^9\right)^{\frac{1}{3}}=x^3.

6. Simplify the expression \sqrt[3]{24 x^4}.

In the expression, \sqrt[3]{24 x^4} look for the factor of 24 that is a perfect cube number. 24 can be written as 8 \times 3.

Expressions with exponents are perfect cubes if the exponent is divisible by 3. So, x^4 can be rewritten as x^3 \times x.

\sqrt[3]{8 \times 3 \times x^3 \times x} , this is the same as \sqrt[3]{8} \; \sqrt[3]{3} \; \sqrt[3]{x^3} \; \sqrt[3]{x} where 8 and x^3 are perfect cube expressions.

So, \sqrt[3]{8}=2, \sqrt[3]{x^3}=x^1 . The other two expressions are not perfect cubes, so they remain under the cube root symbol.

You can also use laws of exponents to simplify the answer.

Cube numbers and cube roots FAQs

The cube root formula can be used to represent any number in the form of its cube root. For example, for a number x, the cube root is represented by \sqrt[3]{x}=(x)^{\frac{1}{3}}, where x^3=x \times x \times x.

Yes, you can find the cube root of any number, even complex numbers.

The prime factorization method can help when finding the cube root of the original number. However, knowing perfect cube numbers can be more helpful.

Yes you can find any root of a number, (the nth root). You might need a calculating device to figure it out.

The next lessons are

• Scientific notation
• Math formulas
• Quadratic graphs

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Cube root of number is a value which when multiplied by itself thrice or three times produces the original value. For example, the cube root of 27, denoted as 3 √27, is 3, because when we multiply 3 by itself three times we get 3 x 3 x 3 = 27 = 3 3 . So, we can say, the cube root gives the value which is basically cubed. Here, 27 is said to be a perfect cube . From the word, cube root, we can understand what is the root of the cube. It means which number caused the cube present under the root. Usually, to find the cubic root of perfect cubes, we use the prime factorisation method. In a similar manner, we can learn the significance of square root here.

In three-dimensional geometry, when we learn about different solids, the cube defines an object which has all its faces or sides equal in dimensions. Also, the formula to find the volume of the cube is equal to side 3 . Hence, if we know the volume of the cube we can easily find the side length of it using cube root formula. This is one of the major applications of cube roots. It defines the cubic root of volume of the cube is equal to the side of it.

Cube Root Symbol

The cube root symbol is denoted by ‘ 3 √’. In the case of square root, we have used just the root symbol such as ‘√’, which is also called a radical. Hence, symbolically we can represent the cube root of different numbers as: Cube root of 5 = 3 √5 Cube root of 11 = 3 √11 And so on. Also, read:

• Cube Root Of Unity
• Important Questions Class 8 Maths Chapter 7 Cubes Roots

Cube Root Formula

As we already know, the cube root gives a value which can be cubed to get the original value. Suppose, cube root of ‘a’ gives a value ‘b’, such that; 3 √a = b This formula is only possible if and only if; a = b 3 This formula is useful when we find the cubic root of perfect cubes. Perfect Cubes:

Taking the cube root of the above cubes is very easy. But for imperfect cubes, we cannot evaluate the cube root this easily. Hence, below is the table for different values of cube root, which students can memorise to solve the questions based on this concept.

Cube Root 1 to 30

How to Find Cube Root?

To find the cubic root of a number easily, we can use the prime factorisation method. By evaluating the prime factors we can pair similar digits in a group of three and take them out as a single digit from the cubic root. Let us take an example of finding the cube root of 8. By factoring 8, we get; 8 = 2 x 2 x 2 8 = 2 3 Now, if we take the cubic root both the sides, then the cube of 2 cancels the cubic root. Hence, we get the value of 3 √8 Therefore, 3 √8 = 2 Also, check: How to find cube root by Prime factorisation and Estimation Method

How to Simplify Algebraic Cube Roots?

To simplify the algebraic cubic roots, the cubic radical should possess the following conditions:

• Under the radical symbol, there should be no fractional value
• There should be no perfect power factors under the cube root symbol
• Under the cube root symbol, no exponent value should be greater than the index value.
• If the fraction is appearing under the radical, the denominator of the fraction should not have any fraction.

While finding the cube root of any number, we will search for the factors which occur in the set of three. For example, the cube of 8 is 2. The factor of 8 is 2 x 2 x 2. 

Unlike square roots, cube roots should not be concerned with the negative values under the radical sign. Thus, perfect cubes can also possess negative values. It should be noted that perfect squares cannot be a negative value.

For example, a cube root of – 125 is -5.

Because 125 is a perfect cube, as 125 = 5 x 5 x 5

While dealing with the perfect cube numbers, separate and determine the perfect cube factors. Sometimes, it might be helpful if we separate the expression under the radical symbol. But be sure to write the cube root for each section. Also, check each section will possess the property of perfect cubes. 

Cube Root Questions

1. Find the cube root of 64. Solution: To find the cube root of 64, we need to use the prime factorisation method . 64 = 2×2×2×2×2×2 64 = 4 × 4 × 4 64 = 4 3 Now taking the cube root on both the sides, we get; 3 √64 = 3 √(4 3 ) 3 √64 = 4 2. What is the cube root of 1331? Solution: Using the prime factorisation method, we get; 1331 = 11×11×11 1331 = 11 3 3 √1331 = 11 3. Find the cube root of 216. Solution: By prime factorisation, we know; 216 = 2×2×2×3×3×3 216 = 2 3 ×3 3 216 = (2×3) 3 = 6 3 3 √216 = 6 4. Find 3 √343. Solution: By prime factorisation 343 = 7x7x7 343 = 7 3 3 √343 = 7 5. Evaluate the value of 3 √1728. Solution: Using prime factorisation method; 1728 = 2×2×2×2×2×2×3×3×3 1728 = 2 3 ×2 3 x3 3 1728 = (2×2×3) 3 1728 = 12 3 3 √1728 = 12

Video Lesson

Frequently Asked Questions on Cube Roots

Define cube root.

In Mathematics, the cube root of a number “a” is a number “b”, such that b3 = a. It means that the cube root of a number gives a value which when cubed gives the original number.

Can we find the cube root for negative numbers?

Yes, we can find the cube root of a negative number. For example, the cube root of -64 is -4.

What is the cube root of 512?

The cube root of 512 is 8 because 512 is a perfect cube. When 8 is multiplied thrice, we get 512.

What is the difference between the square root and cube root?

A cube root is a number, which when cubed gives the radicand, whereas the square root is a number which when squared gives the radicand. Also, the cube root of a negative number can be negative whereas the square root of a negative number cannot be negative.

How to find the cube root of a number?

The cube root of a number can be found using the prime factorization method or the long division method.

Learn More :

• Cube Root of 2
• Cube Root of 4
• Cube Root of 64
• Cube Root of 216
• Cube Root of 343
• Cube Root of 512
• Cube Root of 729
• Cube Root of 1728
• Cube Root of 2197
• Cube Root of 9261

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Cube Root Calculator

Table of contents

Our cube root calculator is a handy tool that will help you determine the cube root, also called the 3 rd root, of any positive number . You can immediately use our calculator; just type the number you want to find the cube root of and it's done! Moreover, you can do the calculations the other way around and use them to cube numbers. To do this just type the number you want to raise to third power in the last field! It may be extremely useful while searching for so-called perfect cubes . You can read more about them in the following article.

Thanks to our cube root calculator, you may also calculate the roots of other degrees . To do so, you need to change the number in the degree of the root field. If you would like to learn more about the cube root definition, familiarize yourself with the properties of the cube root function, and find a list of the perfect cubes, we strongly recommend you keep on reading this text. In there, you can also find some tricks on how to find the cube root on a calculator or how to calculate it in your head.

If you are interested in the history of root symbols head to the square root calculator , where we discuss it.

Cube root definition

Let's assume you want to find the cube root of a number, x . The cube root, y , is such a number that, if raised to the third power, will give x as a result. If you formulate this mathematically,

∛x = y ⟺ y^3 = x

where ⟺ is a mathematical symbol that means if and only if .

It is also possible to write the cube root in a different way, which is sometimes much more convenient. It is because a cube root is a special case of an exponent. It can be written down as

∛(x) = x^(1/3)

A geometric example may help you understand this. The best example we can give would be that of the cube. Well, the cube root of a cube's volume is its edge length. So, for example, if a cube has a volume of 27 cm³, then the length of its edges is equal to the cube root of 27 cm³, which is 3 cm. Easy?

You should remember that in most cases, the cube root will not be a rational number . These numbers can be expressed as a quotient of two natural numbers, i.e., a fraction. Fractions can cause some difficulties, especially when it comes to adding them. If you are having trouble working with fractions, try our adding fractions calculator , which will help you immensely.

What is the cube root of...?

It is really easy to find the cube root of any positive number with our cube root calculator! Simply type in any number to find its cube root. For example, the cube root of 216 is 6. For the list of perfect cubes, head to the next section .

Note that it is possible to find a cube root of a negative number as well. After all, a negative number raised to the third power is still negative - for instance, (-6)³ = -216 .

You need to remember, though, that any non-zero number has three cube roots: at least one real one and two imaginary ones. This cube root calculator deals with real numbers only, but if you're interested, we encourage you to read more on the topic of imaginary numbers!

Most common values - perfect cubes list

You can find the most common cube root values below. Those numbers are also very often called perfect cubes because their cube roots are integers. Here is the list of the ten first perfect cubes:

• cube root of 1: ∛1 = 1 , since 1 * 1 * 1 = 1 ;
• cube root of 8: ∛8 = 2 , since 2 * 2 * 2 = 8 ;
• cube root of 27: ∛27 = 3 , since 3 * 3 * 3 = 27 ;
• cube root of 64: ∛64 = 4 , since 4 * 4 * 4 = 64 ;
• cube root of 125: ∛125 = 5 , since 5 * 5 * 5 = 125 ;
• cube root of 216: ∛216 = 6 , since 6 * 6 * 6 = 216 ;
• cube root of 343: ∛343 = 7 , since 7 * 7 * 7 = 343 ;
• cube root of 512: ∛512 = 8 , since 8 * 8 * 8 = 512 ;
• cube root of 729: ∛729 = 9 , since 9 * 9 * 9 = 729 ;
• cube root of 1000: ∛1000 = 10 , since 10 * 10 * 10 = 1000 ;

As you can see, numbers become very large quickly, but sometimes you'll have to deal with even bigger numbers, such as factorials. In this case, we recommend using scientific notation, which is a much more convenient way of writing down really big or really small numbers.

On the other hand, most other numbers are not perfect cubes , but some of them are still used often. Here is the list of some of the non-perfect cubes, rounded to the hundredths:

• cube root of 2: ∛2 ≈ 1.26 ;
• cube root of 3: ∛3 ≈ 1.44 ;
• cube root of 4: ∛4 ≈ 1.59 ;
• cube root of 5: ∛5 ≈ 1.71 ;
• cube root of 10: ∛10 ≈ 2.15 ;

Don't hesitate to use our cube root calculator if the number you want and need is not on this list!

Cube root function and graph

You can graph the function y = ∛(x) . Unlike e.g. the logarithmic function, the cube root function is an odd function - it means that it is symmetric with respect to the origin and fulfills the condition - f(x) = f(-x) . This function also passes through zero.

Thanks to this function, you can draw a cube root graph, which is shown below. We also encourage you to check out the quadratic formula calculator to look at other function formulas!

How to calculate cube root in your head?

Do you think that it is possible to solve simple problems with cube roots without an online calculator, or even a pencil or paper? If you think that it is impossible, or that you are incapable of doing it check out this method, it is very easy. However, it only works for perfect cubes . Forget all the rules in the arithmetic books and consider for a moment the following method described by Robert Kelly.

First of all, it is essential to memorize the cubes of the numbers from 1 to 10 and the last digit of their cubes. It is presented in the table below.

When you have a number you want to find the cube root of look first at the thousands (skip the last three digits). For example, for the number 185,193 , The thousands are 185. The cube of 5 is 125 and of 6 is 216. Therefore it is obvious that the number you are searching for is between 50 and 60. The next step is to ignore all the other figures except the last digit. We can see that it's 3, so check your memory or in our table. You will find that the number you are searching for is 7. So the answer is 57 ! Easy?

Let's take another example and do it step by step!

• Think of the number that you want to know as a cube root. Let's take 17576 .
• Skip the three last digits.
• Find the two closest cube roots that you know. The cube root of 8 is 2, and the cube root of 27 is 3. So your number is between 20 and 30.
• Look at the last digit. The last digit of 17576 is 6.
• Check your memory (or on our table) - the last digit 6 corresponds with the number 6. This is the last digit of your number.
• Combine the two: 26 . This is the cube root of 17576!

We remind you that this algorithm works only for perfect cubes! And the probability that a random number is a perfect cube is, alas, really low. You've got only a 0.0091 percent chance of finding one between 1,000 and 1,000,000. If you're not sure about your number, just forget about that rule and use our cube root calculator :-)

How do I find the cube root on a regular calculator?

• First, you need to type the number for which you need to find the cube root
• Press √ (root key) two times
• Press x (multiplication sign)
• Press √ (root key) four times
• Press √ (root key) eight times
• One last time, press the √ (root key) two times
• And now you can press = (equal to sign)! Here is your answer!

Don't you believe it? Check it one more time with another example!

Examples of cube root questions

Let's say you need to make a ball with a volume of 33.5 ml. To prepare it you need to know its radius. As you probably know, the equation for calculating the volume of a sphere is as follows:

V = (4/3) * π * r³

So the equation for the radius looks like this:

r = ∛(3V/4π)

You know that the volume is 33.5 ml. At first, you need to switch to different volume units. The simplest conversion is into cm³: 33.5 ml = 33.5 cm³. Now you can solve the radius:

r = ∛(100.5/12.56)

For a ball to have a volume of 33.5 ml, its radius should be 2 centimeters.

nth root calculator

With our root calculator, you can also calculate other roots. Just write the number in the Degree of the root field, and you will receive any chosen nth root calculator . Our calculator will automatically do all necessary calculations, and you can freely use it in your calculations!

So, let's take some examples. Let's assume you need to calculate the fourth root of 1296 . First, you need to write the appropriate number you want to root - 1296. Then change the degree of the root to 4 . And you've got the result! The fourth root of 1296 is 6 .

Our nth root calculator also enables you to calculate the root of irrational numbers. Let's try it by calculating π-th root. Symbol π represents the ratio of a circle's circumference to its diameter. Its value is constant for every circle and is approximately 3.14, but you can use our ratio calculator to find its more precise value!

Let's say you want to calculate the π-th root of 450 . First, write 450 in the number box. Then change the degree of the root - let's round and write 3.14 instead of π. And now you can see the result. It's almost 7 .

Three solutions of the cube root

At the end of this article, we've prepared an advanced mathematics section for the most persistent of you. You probably know that positive numbers always have two square roots: one negative and one positive. For example, √4 = -2 and √4 = 2 . But did you know that a similar rule applies to the cube roots? All real numbers (except zero) have exactly three cube roots : one real number and a pair of complex ones. Complex numbers were introduced by mathematicians a long time ago to explain problems that real numbers cannot do. We usually express them in the following form:

x = a + b*i

where x is the complex number with the real a and imaginary b parts (for real numbers b = 0 ). The mysterious imaginary number i is defined as the square root of -1 :

Alright, but how does this knowledge influence the number of cube root solutions? As an example, consider the cube roots of 8 , which are 2 , -1 + i√3 , and -1 - i√3 . If you don't believe us, let's check it by raising them to the power of 3, remembering that i² = -1 and using the short multiplication formula (a + b)³ = a³ + 3a²b + 3ab² + b³ :

• 2³ = 8 - the obvious one,
• (-1 + i√3)³ = -1 + 3i√3 + 9 - 3i√3 = 8 ,
• (-1 - i√3)³ = -1 - 3i√3 + 9 + 3i√3 = 8 .

Do you see it now? All of them equal 8 !

How do I find the cube root of a product?

The cube root of a product of two numbers is the product of the cube roots of these numbers. That is, the formula is ∛(a × b) = ∛a × ∛b .

What is the cube root of -8/27?

The answer is -2/3 . To get this result, take these steps:

• Recall the formula ∛(a / b) = ∛a / ∛b .
• Compute the cube root of -8 . Clearly, ∛(-8) = -2 .
• Compute the cube root of 27 : we have ∛27 = 3 .
• The final result is -2/3 . Well done!

How do I write the cube root on a computer?

The Alt code for the cube root ∛ symbol is 8731 . That is, to produce ∛, take these steps:

• Make sure the Num Lock is on.
• Press down one of the Alt keys .
• Holding down the Alt key, type the code 8731 using the numeric keypad .
• Let go of the Alt key. The cube root symbol will appear.
• Alternative method: copy the ∛ symbol (Ctrl+C) and paste it wherever you need it (Ctrl+V).

Degree of the root

Cubes and Cube Roots

To understand cube roots, first we must understand cubes ...

How to Cube A Number

To cube a number, just use it in a multiplication 3 times ...

Example: What is 3 Cubed?

Note: we write "3 Cubed" as 3 3 (the little 3 means the number appears three times in multiplying)

Cubes From 0 3 to 6 3

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

The cube root of a number is ... ... a special value that when cubed gives the original number.

The cube root of 27 is ... ... 3 , because when 3 is cubed you get 27 .

Here are some cubes and cube roots:

Example: What is the Cube root of 125?

Well, we just happen to know that 125 = 5 × 5 × 5 (if we use 5 three times in a multiplication we get 125) ...

... so the cube root of 125 is 5

The Cube Root Symbol

You can use it like this:

You Can Also Cube Negative Numbers

Have a look at this:

So the cube root of −125 is −5

Perfect Cubes

The Perfect Cubes are the cubes of the whole numbers :

It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

Example: what is the cube root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

• Let's try 3.5: 3.5 × 3.5 × 3.5 = 42.875
• Let's try 3.2: 3.2 × 3.2 × 3.2 = 32.768
• Let's try 3.1: 3.1 × 3.1 × 3.1 = 29.791

We are getting closer, but very slowly ... at this point, I get out my calculator and it says:

3.1072325059538588668776624275224 ...

... but the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation ! (Further reading: these kind of numbers are called surds which are a special type of irrational number )

Cube Root Calculator

Use this cube root calculator to easily calculate the cube root of a given number.

Related calculators

• What is a cube root?
• Properties and practical application of the cube root
• Commonly used cube roots
• Does the calculator support fractions?

    What is a cube root?

The cube root of a number answers the question "what number can I multiply by itself twice to get this number?". It is the reverse of the exponentiation operation with an exponent of 3, so if r 3 = x, then we say that "r is the cube root of x". Finding the root of a number has a special notation called the radical symbol: √ - which is used as it is when it comes to square roots, but gets an index number indicating the root - 3 in the case of the cube root, so it looks like so: ∛. Usually the radical spans over the entire equation for which the root is to be found. It is called a "cube" root since multiplying a number by itself twice is how one finds the volume of a cube .

Unlike the square root, there is only one unique real number root as a result from applying the cube root function for a given number and it carries the sign of the number. For example, the ∛8 = 2, while ∛-8 = -2, since 2 x 2 x 2 = 8 and -2 x -2 x -2 = -8. The real number cube root is the Principal cube root, but each real number cube root (zero excluded) also has a pair of complex conjugate roots. For example, the other cube roots of 8 are -1 + √3 i and -1 - √3 i . Our cube root calculator will only output the principal root. See the table of common roots below for more examples.

When interpreting the output of the calculator it might help to know that in geometrical terms, the cube root function maps the volume of a cube onto its side length.

    Properties and practical application of the cube root

Cubic roots are useful when one needs to divide an angle in three, that is - to find an angle whose measure is one third of a given angle. This operation is called angle trisection. Cube roots are used when you need to find the edge of a cube whose volume is twice that of a cube with a given edge.

Plotting the results from the cube root function, as calculated using this calculator, on a graph reveals that it has the shape of half a parabola.

    Commonly used cube roots

Table of commonly encountered cube roots:

The calculations were performed using this cube root calculator.

    Does the calculator support fractions?

Yes, simply enter the fraction as a decimal floating point number and you will get the corresponding cube root. For example, to compute the cube root of 1/2 simply enter 0.5 in the input field and you will get 0.7937 as ouput. If you are having trouble converting a fraction to a decimal number, you will find our fraction to decimal converter handy.

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Cube Root Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/cube-root-calculator.php URL [Accessed Date: 07 Sep, 2024].

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Cube Root Calculator

Cube root calculator finds the principal (real) cube root of positive and negative numbers and the imaginary cube roots of the given number.

Positive or Negative Number

Related Calculators

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Table of Contents

Directions for use, cube root definition, perfect cubes, cube root properties, calculating the real cube root of a perfect cube, calculating the real cube root of a number greater than -1 and less than 1 (excluding 0), the cubic volume of wood.

This calculator can be used for finding all cube roots of the given number. It finds both real and imaginary roots.

To find the cube root of a number, enter that number into the input field and press "Calculate." The calculator will demonstrate the answer in two parts: the "principal (real) root", and "all roots," where "all roots" include the principal root and the imaginary roots.

The calculator accepts positive and negative integers as inputs. Fractions and imaginary numbers are not accepted. Note that if you use a fraction or an imaginary number as an input, this cube roots calculator will automatically disregard everything following the first non-number symbol. For example, if you enter 8/15, the calculator will calculate the cube root of 8; if you enter 5 + 3i, the cube root of 5 will be calculated.

The cube root of a number is defined as the number that has to be multiplied three times to get the original number. The cube root of x is commonly denoted as ∛x. According to the definition, y is the cube root of x:

$$y=\sqrt[3]{x}$$

$$y \times y \times y = x$$

Taking a cube root of a number, ∛x, is equivalent to raising that number to the power of 1/3:

$$\sqrt[3]{x}=x^{\frac{1}{3}}$$

Cube root operation is the reverse of finding the cube operation. To find the cube of a number, that number has to be multiplied 3 times:

$$y^3 = y \times y \times y = x$$

And inversely,

$$\sqrt[3]{x}=\sqrt[3]{y×y×y}=y$$

A perfect cube is a number, the cube root of which is an integer. For example, 8 is a perfect cube since:

$$\sqrt[3]{8}=\sqrt[3]{2×2×2}=2$$

Since integers are whole numbers that can be positive and negative, perfect cubes can be both positive and negative. For example, -8 is a perfect cube since:

$$\sqrt[3]{-8}=\sqrt[3]{-2×-2×-2}=-2$$

0 is also an integer and

$$\sqrt[3]{0}=\sqrt[3]{0×0×0}=0$$

Therefore, 0 is also a perfect cube.

On the other hand, 4 is not a perfect cube since the real cube root of 4:

∛4 ≈ 1.58740105

which is not an integer.

A cube root of a negative number is defined as the negative of the cube root of a positive number, i.e.,

$$\sqrt[3]{-x}=-\sqrt[3]{x}$$

For example,

$$\sqrt[3]{-27}=-\sqrt[3]{27}=-3$$

Multiplication property of cube roots:

$$\sqrt[3]{x}×\sqrt[3]{y} =\sqrt[3]{x×y}$$

How to calculate cube root

To find the cube root of a number, use the prime factorization method:

• Find prime factors of the number.
• Divide prime factors into groups containing three factors that are the same.
• Take one factor of each of the groups, and multiply them to get the final answer.

For example, let's find all the real cube roots of 3375, ∛3375:

• Finding prime factors of 3375, we get 3375 = 3 × 3 × 3 × 5 × 5 × 5.
• Dividing them into groups of three same factors, we get 3375 = (3 × 3 × 3) × ( 5 × 5 × 5).
• Finally, taking one factor of each group and multiplying them, we get 3 × 5 = 15.

Therefore, ∛3375 = 15.

If the prime factors of a number do not form groups of three, the number is not a perfect cube, and we cannot use this method to find the cube root.

If the given number is greater than -1 and less than 1, it cannot be a perfect cube since by definition, a perfect cube is a number, the cube root of which is an integer. Any number y from the interval -1 < y < 1 that is not 0 cannot be a perfect cube. However, sometimes finding the real cube root of such a number can be relatively easy.

For example, let's find all the real cube roots of -0.000125. This number is not an integer. Therefore, we cannot use the prime factorization method described above.

But we can easily notice that -0.000125 = -125 × 10⁻⁶. Therefore,

$$\sqrt[3]{-0.000125}=\sqrt[3]{(-125)×10⁻⁶}$$

Applying the multiplication property of the cube root, we get:

$$\sqrt[3]{-0.000125}=\sqrt[3]{(-125)×10⁻⁶}=\sqrt[3]{(-125)}×\sqrt[3]{10⁻⁶}$$

Rewriting the cube root of the negative number as the negative of the cube root of the positive number, we get:

$$\sqrt[3]{(-125)}×\sqrt[3]{10⁻⁶}=-\sqrt[3]{(125)}×\sqrt[3]{10⁻⁶}$$

It is easy to notice that 125 = 5 × 5 × 5, and 10⁻⁶ = 10⁻² × 10⁻² × 10⁻². Therefore,

$$\sqrt[3]{(125)}=\sqrt[3]{(5×5×5)}=5$$

$$\sqrt[3]{(10⁻⁶)}=\sqrt[3]{(10⁻²)×(10⁻²)×(10⁻²)=10⁻²}$$

Finally, we get:

$$\sqrt[3]{(-0.000125)}=\sqrt[3]{((-125) × 10⁻⁶)}=\sqrt[3]{(-125)}×\sqrt[3]{(10⁻⁶)}$$

$$\sqrt[3]{(-125)}×\sqrt[3]{(10⁻⁶)}=-\sqrt[3]{(125)}×\sqrt[3]{(10⁻⁶)}$$

$$-\sqrt[3]{(125)}×\sqrt[3]{(10⁻⁶)}=-\sqrt[3]{(5×5×5)}×\sqrt[3]{(10⁻²)×(10⁻²)×(10⁻²)}=(-5)×10⁻²=-0.05$$

Real life examples

Cube roots are used in real life to find the side length of any cubic object. For example, if you know a box's volume and want to find how high it is, check whether it would fit somewhere. Or, if you need to estimate the amount of paint, you would need to paint the walls of a cubic room. Or, if you need to count the number of tiles, you need to cover the floor of a cubic room with a known volume.

Imagine building a house and finding an ad for 64 cubic meters of wood for sale. What would the dimensions of that volume of wood be in length, width, and height?

To solve this problem, you must find the cube root of 64. The length of the side of the imaginary cube that would help you describe this volume would be ∛64 = 4. Thus, from the original data on the cubic volume of wood, we have a different idea of the size of such a volume.

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How to Calculate Cube Root by Hand

Last Updated: September 7, 2024

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 1,307,532 times.

With the use of calculators, finding the cube root of any number may be just buttons away. But perhaps you don't have a calculator, or you want to impress your friends with the ability to calculate a cube root by hand. There is a process that appears a bit laborious at first, but with practice it works fairly easily. It is helpful if you remember some basic math skills and some algebra about cube numbers.

Working Through a Sample Cube Root Problem

• Write down the number whose cube root you want to find. Write the digits in groups of three, using the decimal point as your starting place. For this example, you will find the cube root of 10. Write this as 10. 000 000. The extra 0s are to allow precision in the solution.
• Draw a cube root radical sign over the number. This serves the same purpose as the long division bar line. The only difference is the shape of the symbol.
• Place a decimal point above the bar line, directly above the decimal point in the original number.

• In this example, the first set of three numbers is 10. Find the largest perfect cube that is less than 10. That number is 8, and its cube root is 2.

• To the left of the vertical line, you will be solving the next divisor, as the sum of three separate numbers. Draw the spaces for these numbers by making three blank underlines, with plus symbols between them.

• Now calculate 3 times 10 times each of the two digits that are in your solution above the radical sign. For this sample problem, that means 3*10*2*1, which is 60. Add this to the 1200 that you already have to make 1260.
• Finally, add the square of the last digit. For this example, that is a 1, and 1^2 is still 1. The total divisor is, therefore 1200+60+1, or 1261. Write this to the left of the vertical line.

• You can check the precision of this result by cubing 2.1*2.1*2.1. The result is 9.261.
• If you believe your result is precise enough, you can quit. If you want a more precise answer, then you need to proceed with another round.

• Drop down the next group of three digits. In this case, these are three 0s, which will follow the 739 remainder to give 739,000.

• Select the next digit of your solution so that you can multiply it by 132,300 and have less than the 739,000 of your remainder. A good choice would be 5, since 5*132,300=661,500. Write the digit 5 in the next space above the radical line.

• Add the parts of your divisor to get 132,300+3,150+25=135,475.

• Multiply the divisor by the last digit of your solution. 135475*5=677,375.
• Subtract. 739,000-677,375=61,625.

Finding Cube Roots by Repeated Estimation

• In the working example, the target of 600 falls about halfway between the boundary numbers of 512 and 729. So, select 5 for your next digit.

Understanding How this Calculation Works

• For more about expanding the binomial to get this result, read about how to multiply binomials . For a more advanced, shortcut version, look for the information about calculating (x+y)^n with Pascal's Triangle.

• The first term contains a multiple of 1000. You first a number that could be cubed and stay within the range for the long division for the first digit. This provides the term 1000A^3 in the binomial expansion.

• The second digit in each step of the cube root calculation comes from the third term of the binomial expansion. In the binomial expansion, you can see the term 30AB^2.
• The final digit of each step is the term B^3.

Calculator, Practice Problems, and Answers

Community Q&A

• As with everything else in mathematics, practice makes perfect. The more you practice, the better you will get at this calculation. Thanks Helpful 0 Not Helpful 0

• It is easy to make a calculation error. Check your work carefully and review. Thanks Helpful 3 Not Helpful 3

Things You'll Need

• Pen or pencil
• Piece of paper

You Might Also Like

• ↑ https://youtu.be/y0qWHMmCY4E?t=60
• ↑ https://www.had2know.org/academics/compute-cube-root-by-hand.html
• ↑ https://youtu.be/y0qWHMmCY4E?t=230
• ↑ https://youtu.be/y0qWHMmCY4E?t=260
• ↑ https://www.themathdoctors.org/evaluating-square-roots-by-hand/
• ↑ https://xlinux.nist.gov/dads/HTML/cubeRoot.html

About This Article

To calculate cube root by hand, choose a perfect cube that is as close to the answer as possible, write it down, and subtract your estimate from the original number. For example, you could estimate that the square root of 30 was 3. However, 3 cubed is 27, so you would write down 3 as the first part of your answer with a remainder of 3. Then, estimate what cubed would fit into the remainder and subtract it, too. Repeat that process until you’ve reached your desired accuracy. Keep reading to learn how to find cube roots through long division. Did this summary help you? Yes No

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Home » Tech Tips » Shortcuts » How to Type Square Root, Cube Root and Fourth Root Symbols?

How to Type Square Root, Cube Root and Fourth Root Symbols?

It is easy to tell square root of 9 is equal to 3. However, typing this in a readable format is not simple. You can use equation editor in Microsoft Office or use dedicated applications like LaTex. Unfortunately, using equation editor in Word or Excel is difficult task as the content will not align with other text content on your document. Also, for the use of few times you don’t need equation editor.  When you want to type square root, cube root and fourth root symbols on your documents then the easy way is to use alt code shortcuts .

Alt Code Shortcut for Square Root Symbol

Square root or principal square root symbol √ does not have 2 on the root. When A 2= B then A is the square root of B indicated as √B = A. For example, √4 = 2. Here are the shortcuts for square root in Windows and Mac computers.

Alt Code Shortcut for Cube Root Symbol

When A­­ 3 = B then A is the cube root of B indicated as ∛B = A. For example, ∛8 = 2.

Alt Code Shortcuts for Fourth Root Symbol

When A 4 = B then A is the fourth root of B indicated as ∜B = A. For example, ∜16 = 2.

Typing Square Root, Cube Root and Fourth Root in Windows

Follow one of the methods in Windows based documents like Word, PowerPoint, Excel and Outlook.

• Press the alt key and type 8730 using numeric keypad to make square root √ symbol.
• Only on Microsoft Word documents, type 221B and press alt and x keys to make cube root symbol ∛.
• Press “Win + ;” keys to open Windows emoji keyboard. Click on the “Symbols” icon and then “Math” symbols. Search and insert square root and other high-order root symbols.
• When you are in Office application like Word, go to “Insert > Symbols” and insert root symbols.
• Enable Math AutoCorrect function to type with keyboard shortcuts like below:

Inserting on Mac Documents

• On MacBook, change the keyboard layout to Unicode Hex Input . Hold the option or alt key and type 221C to produce fourth root symbol ∜.
• Open Character Viewer by pressing “Control + Command + Space”. Either go to “Math Symbols” or search for “root” using the search box. Find the square, cube or fourth root symbols to insert on Pages, Keynote and Numbers.

Symbol Display

As you can see the root symbols will not have the top horizontal line when typing with shortcuts. However, on Mac , you can select the font variations from Character Viewer that includes the top bar in the root symbols.

Other Names for Root Symbols

In mathematics , square root and other root symbols are referred with the below names.

• Radical symbol
• Radical sign
• Root symbol

Arabic Cube and Fourth Root Symbols

The Unicode system has two more root symbols in Arabic as listed below.

About Editorial Staff

Editorial Staff at WebNots are team of experts who love to build websites, find tech hacks and share the learning with community.

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Leave your reply.

How about the nth root symbol? How do I type that?

Thanks for the article but the cube root of 9 is not 3 and the fourth root of 16 is not 4. Maybe stick with powers of 2 for the examples. The cube root of 8 is 2 and the fourth root of 16 is 2 as well.

Thanks for the inputs, we have corrected the mistakes.

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Cubes and Cube Roots (A)

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Proving a Factor of Complex Cube Root of 1 in x^3 + y^3 + z^3 - 3xyz Equation

• Thread starter Ferrus
• Start date Feb 5, 2009
• Tags Complex Cube Root
• Feb 5, 2009

Homework Statement

Homework equations, the attempt at a solution.

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Ferrus said: If w is a complex cube root of 1, prove that x + wy + w^2z is a factor of x^3+ y^3 + z^3 - 3xyz …

FAQ: Proving a Factor of Complex Cube Root of 1 in x^3 + y^3 + z^3 - 3xyz Equation

What is the complex cube root of 1.

The complex cube root of 1 is a mathematical concept that refers to the three complex numbers that, when multiplied together, result in 1. These numbers are known as the cube roots of 1.

What are the three solutions to the complex cube root of 1?

The three solutions to the complex cube root of 1 are 1, (-1 + √3i)/2, and (-1 - √3i)/2. These three numbers, when multiplied together, equal 1.

How do you calculate the complex cube root of 1?

The complex cube root of 1 can be calculated by using the formula: 1^(1/3) = e^(2πki/3), where k = 0, 1, 2. This formula will give the three solutions mentioned above.

What is the relationship between the complex cube root of 1 and the unit circle?

The three solutions to the complex cube root of 1 can be represented on the unit circle, with each solution corresponding to a point on the circle. This is because the solutions are complex numbers in the form of a + bi, where a and b represent the coordinates on the unit circle.

Why is the complex cube root of 1 important in mathematics?

The complex cube root of 1 is important in mathematics because it is a fundamental concept in understanding complex numbers and their properties. It is also used in various mathematical equations and applications, such as in engineering and physics.

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What is the cube root of 64?

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