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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.

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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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Simplifying radicals

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.

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.

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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 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

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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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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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How to Put a Cube Root Into a Graphing Calculator

Graphing Calculator Art: How to Make a Face

With a little practice, you may get pretty good at spotting the cube roots of easy numbers. For example, 3 √8 = 2, 3 √27 = 3 and so on. But when it comes to finding cube roots for larger numbers, or finding exact values for cube roots that don't work out to a whole number, a scientific calculator becomes a very useful tool. If you're using a calculator with graphing capability, you can also access a graph of this function.

Finding a Cube Root on a TI-83/84 Calculator

The TI-83/84 series of calculators is the most popular graphing calculator you may encounter in academic settings, and all models use the same process to access cube roots.

Press the MATH key, located on the far left side of the calculator, to bring up a menu of special operations.

Press 4 to select the cube root function, then input the number you'd like to find the cube root of and press ENTER. The calculator will return the value of the cube root.

Graphing a Cube Root on a TI-83/84 Calculator

Again, all versions of the TI-83/84 graphing calculator use a similar process to generate a graph of the cube root function.

Press the y = key, located in the top left corner of the calculator, to access the graphing menu.

Press MATH to bring up the menu of special operations, then press 4 to select the cube root function. Next, press the " X, T, θ, n " key, located to the left of the arrow keypad, which generates an x under the cube root function. (In other words, you're asking the calculator to graph 3 √ x .)

Press the GRAPH key, located in the upper right corner of the calculator. This generates a graph of the cube root function.

Finding a Cube Root on a Casio FX Graphing Calculator

Another very popular graphing calculator, the Casio FX series (which includes the FX-9860GII and the FX-9750GII), allows you to access the cube root function straight from the main keypad.

Press the SHIFT key, followed by the ( key. This activates the cube root function.

Enter the number you want to find the cube root of, then hit EXE (execute) to return the result.

Graphing a Cube Root on a Casio FX Graphing Calculator

You can also use the Casio FX series' graphing capability to show a graph of the cube root function.

Press the MENU key, then use the arrow keys to navigate to GRAPH mode. Press EXE to enter graph mode.

Enter the cube root function as just described, with one small difference: Press SHIFT, followed by the ( key, to create the cube root function. Then press the " x , θ, T " key, located on the far left side of the calculator keypad, to enter an x under the cube root sign.

Press F6 to generate the graph of the cube root function.

When You Might Use Cube Roots

The most obvious place you'll use this sort of calculation is in algebra problems. For example, if you're given the equation x 3 = 125, you'll need to use the cube root function to solve for x . In the real world, cube roots pop up when you consider problems in three dimensions or, to put it another way, when you start calculating volume.

For example, if you're trying to figure out the dimensions of a square-shaped container whose volume you already know, you could use the cube root function to find the length of its sides. That's because the volume of a square container is y 3 or y × y × y , where y is the length of one of its sides. So if you already know the volume V , you calculating 3 √ V gives you the length of each side.

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• Andrew Borne: Cube Roots on the TI-84 With Graph
• Exeurbia: How to Find the Cubed Root on a TI 83/84 Calculator
• Casio Support: User's Guide for FX-9860GII, FX-9750GII

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Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.

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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 …

Related to 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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To use cube root, do ^(1/3) like:

https://www.mathpapa.com/algebra-calculator.html?q=8%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D