Cube and Cube Roots are fundamental concepts in algebra and it is introduced in the early classes. The multiplication of a number to itself gave rise to a square and then if we multiply the number by its square then the result becomes a cube and the inverse of the cube is the cube root which we will study in this article. In this article, we will learn about cubes and cubes and also learn about the methods to find both cubes of a number and cube roots of a number. So, let’s dive into the concept of Cubes and Cube Roots.

Here is how, this is the best way, I have found:

x = int(raw_input(«Enter an integer: «))

for ans in range(0, abs(x) + 1):

if ans ** 3 == abs(x):

break

if ans ** 3 != abs(x):

print x, ‘is not a perfect cube!’

else:

if x < 0:

ans = -ans

print ‘Cube root of ‘ + str(x) + ‘ is ‘ + str(ans)

Is there a better way, preferably one that avoids having to iterate over candidate values?

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asked Jan 18, 2015 at 20:04

You could use x ** (1. / 3) to compute the (floating-point) cube root of x.

def is_perfect_cube(x):

x = abs(x)

return int(round(x ** (1. / 3))) ** 3 == x

print(is_perfect_cube(63))

print(is_perfect_cube(64))

print(is_perfect_cube(65))

print(is_perfect_cube(-63))

print(is_perfect_cube(-64))

print(is_perfect_cube(-65))

print(is_perfect_cube(2146689000)) # no other currently posted solution

# handles this correctly

This takes the cube root of x, rounds it to the nearest integer, raises to the third power, and finally checks whether the result equals x.

The reason to take the absolute value is to make the code work correctly for negative numbers across Python versions (Python 2 and 3 treat raising negative numbers to fractional powers differently).

answered Jan 18, 2015 at 20:07

The best way is to use simple math

Complete Program for all the requirements as specified

answered Jan 18, 2015 at 20:04

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def cube(x):

if 0<=x: return x**(1./3.)

return -(-x)**(1./3.)

print (cube(8))

print (cube(-8))

Here is the full answer for both negative and positive numbers.

Or here is a one-liner;

root_cube = lambda x: x**(1./3.) if 0<=x else -(-x)**(1./3.)

answered Jan 18, 2015 at 20:23

The cube root of a number is an important concept like square roots in the number system. Note that the cube root is the inverse method of finding cubes. In this section, we will discuss about the cube root of a number.

What is cube and cube root? The number obtained by multiplying a given number two times by itself is called the of that given number. So is the cube of . Here the number is called the cube root of .

- Definition of Cube of a Number
- Definition of Cube Root
- Cube Root Symbol
- Rules of Cube Root
- Simplifying Cube Roots
- Methods of Finding Cube Root
- Addition or Subtraction of Cube Roots
- Multiplication or Division of Cube Roots
- Cube Root of a Decimal
- Cube Root of a Complex Number
- Cube Root as a Function
- Generalization of Cube Roots
- Solved Problems of Cube Roots
- Cube Root of Numbers
- Prime Factorization Method
- Approximation Method
- Sample Problems
- How to Find a Cube Root?
- Cubes and Cube Roots of the First 20 Natural Numbers
- How to Find the Cube Root by Estimation Method?
- FAQs on Cube and Cube Roots
- What is the Cube of 3?
- What is the Cube of 4?
- How do you calculate the cube root of a number?
- What is the Cube Root of 27?
- What is the Cube Root of 125?
- What is the relationship between Cubes and Cube roots?
- How to Find Cube Root of a Number?
- Applications of Cube Root Formula
- Solved Example on Cube Root Calculator
- Representation of Cube Numbers
- What Is a Cube Root Calculator?
- Unit Digits in Cube Numbers
- Cube Root Formula for Negative Numbers
- What is Cube?
- Cube 1 to 20
- Properties of Cube of Numbers
- Cube of Fractions
- List of Cube Root of Numbers
- Cube of a Fraction
- How to Use the Cube Root Calculator?
- FAQs on Cube Root
- What is the Difference Between Cube and Cube Root?
- How to Find the Cube Root of a Number?
- Is it Possible to Simplify the Negative Cube Root?
- What is Not a Perfect Cube?
- What is the Cube of an Odd Natural Number?
- Can the Cube Root of Any Odd Number be Even?
- How to Calculate Cube Root of Any Number?
- What is the Cube Root Formula in Algebra?
- What is the Cube Root Formula for Negative Numbers?
- How to Use Cube Root Formula?
- By Prime Factorization
- Cube Root of a Cube Number using Estimation
- Hardy-Ramanujan Numbers
- What is Cube Root?
- Symbol of Cube Root
- Perfect Cubes
- FAQs on Cube Root Formula
- How To Write Cube Root Formula in Words?
- Cube Root Definition

#### Definition of Cube of a Number

A number is called the cube of a number if the number is a product of three numbers of ‘s. Thus is the cube of .

For example, is the cube of as .

Similarly, is the cube of as .

#### Definition of Cube Root

Thus, if then we say is the cube root of .

Note that , so we can say that is a cube root of .

#### Cube Root Symbol

More examples of cube roots:

#### Rules of Cube Root

Perfect cube table:

Also Read: Fourth root of

#### Simplifying Cube Roots

Let us now learn how to simplify cube roots. We will simplify cube root of .

At first, we will factorize . See that

24 = 2×2×2×3

Taking cube root on both sides, we get

#### Methods of Finding Cube Root

How to find the cube root of a number?

#### Addition or Subtraction of Cube Roots

First, we express the cube roots into their simplified forms as above like cube root of .

Firstly, we will simplify both cube roots.

#### Multiplication or Division of Cube Roots

We write the cube roots into their simplified forms.

#### Cube Root of a Decimal

At first, we need to express the decimal number as a fraction.

Now we will simplify the cube roots of $x$ and $y$ and then we put their values in the fraction obtained in step 2.

Simplifying the fraction obtained in step 3, we will get the cube root of the given decimal number.

To understand the above method, we provide an example here.

Taking square root we get

So the square root of is .

#### Cube Root of a Complex Number

The computation of the cube of a complex number is not as simple as the method of finding the cube root of a real number. We know that the general form of a complex number is $a+ib,$ where both $a$ and $b$ are real numbers.

#### Cube Root as a Function

The above function $f$ is one-to-one but not onto.

#### Generalization of Cube Roots

Let $f(x)$ be a polynomial with a root $c$. Then we must have $f(c)=0.$ This type of roots are classified as polynomial roots.

The computation of cube roots has many applications in several branches of mathematics; such as

#### Solved Problems of Cube Roots

Find the cube root of $125.$

Note that $125=5 imes 5 imes 5$

So the value of the cube root of is .

#### Cube Root of Numbers

I know how to obtain the square root of a number using the sqrt function.

How can I obtain the cube root of a number?

asked Aug 7, 2013 at 12:46

sqrt stands for «square root», and «square root» means raising to the power of 1/2. There is no such thing as «square root with root 2», or «square root with root 3». For other roots, you change the first word; in your case, you are seeking how to perform cube rooting.

Before C++11, there is no specific function for this, but you can go back to first principles:

If you’re expecting to pass negative values for n, avoid the std::pow solution — it doesn’t support negative inputs with fractional exponents, and this is why std::cbrt was added:

std::cout << std::pow(-8, 1/3.) << ‘

‘; // Output: -nan

std::cout << std::cbrt(-8) << ‘

‘; // Output: -2

N.B. That . is really important, because otherwise 1/3 uses integer division and results in 0.

answered Aug 7, 2013 at 12:49

in C++11 std::cbrt was introduced as part of math library, you may refer

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Also, in C++11 there is cbrt in the same header.

Math for Dummies.

answered Aug 7, 2013 at 12:48

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The nth root of x is equal to x^(1/n), so use std::pow. But I don’t see what this has to with operator overloading.

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Just to point this out, though we can use both ways but

long long res = pow(1e9, 1.0/3);

long long res2 = cbrt(1e9);

cout<<res<<endl;

cout<<res2<<endl;

So, in order to get the correct results with pow function we need to add an offset of 0.5 with the actual number or use a double data type i.e.

long long res = pow(1e9+0.5, 1.0/3)

double res = pow(1e9, 1.0/3)

more detailed explanation here C++ pow unusual type conversion

answered Jan 15, 2022 at 11:07

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Actually the round must go for the above solutions to work.

The Correct solution would be

ans = round(pow(n, 1./3.));

answered Jul 31, 2017 at 17:29

The solution for this problem is

cube_root = pow(n,(float)1/3);

Older standards of C/C++ don’t support cbrt() function.

When we write code like cube_root = pow(n,1/3); the compiler thinks 1/3 = 0 (division problem in C/C++), so you need to do typecasting using (float)1/3 in order to get the correct answer

cube root = 4

answered Jul 31, 2019 at 6:27

You can try this C algorithm :

answered May 8, 2022 at 21:27

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I would discourage any of the above methods as they didn’t work for me. I did pow(64, 1/3.) along with pow(64, 1./3.) but the answer I got was 3

Here’s my logic.

answered Jun 28, 2017 at 4:19

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The cube root of a real number x is such a number that when multiplied twice by itself gives us the number x. In other terms, the result generated by multiplying a number x three times is known as the cube of that number. The cube root of the integer x3 is x. The sign ∛x or (x)1/3 represents a number’s cube root. The prime factorization approach and the approximation method are both used to get the cube root of an integer. Let’s take a closer look at both of them.

### Prime Factorization Method

The prime factorization method is used to get the cube root of a perfect cube integer. It is also known as the long division method. A number having an integer cube root is called a perfect cube. It is a number that can be represented as the product of three integers that are all equal. 512, for instance, is a perfect cube since 83 = 8 × 8 × 8 = 512. In this method, the given number is divided by its minimum value factors till the remainder becomes 1. The identical triplets are then grouped together and multiplied to get the value. If any factor cannot be grouped, it is taken inside the cube root which clearly indicates that the given number is not a perfect cube.

We have to find the cube root of 512.

Write the prime factorization of 512.

512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

So, the cube root of the number becomes,

∛512 = ∛ (2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2)

∛512 = 2 x 2 x 2

∛512 = 8

### Approximation Method

This method is used to determine the cube root of non-perfect cubes. It is also known as the estimation method. The important point to be noted while applying this method is that it doesn’t give the exact value of the cube root of a number. Rather, it just provides a value close to the actual cube root. For example, the cube root of 4 is 1.587 but with this method, we get the cube root as 1.67. While applying this method, the prime factorization of the given integer is written first. All prime factors are added and divided by 3 to get an approximate value of the cube root.

We have to estimate the cube root of 3.

Express 3 in the form of its prime factors. So we get,

3 = 1 x 1 x 3

Cube root = (1 + 1 + 3)/3

This provides us the approximate value of cube root of 3, that is, 1.67. However its actual value is 1.442.

### Sample Problems

Problem 1. Calculate the cube root of 64 by the prime factorisation method.

We have to find the cube root of 64.

Write the prime factorization of 64.

64 = 2 x 2 x 2 x 2 x 2 x 2

Taking the cube roots of both sides we get,

∛64 = ∛ (2 x 2 x 2 x 2 x 2 x 2)

Problem 2. Calculate the cube root of 343 by the prime factorisation method.

We have to find the cube root of 343.

343 = 7 x 7 x 7

∛343 = ∛ (7 x 7 x 7)

Problem 3. Calculate the cube root of 125 by the prime factorisation method.

We have to find the cube root of 125.

Write the prime factorization of 125.

125 = 5 x 5 x 5

∛125 = ∛ (5 x 5 x 5)

Problem 4. Calculate the cube root of 729 by the prime factorisation method.

We have to find the cube root of 729.

Write the prime factorization of 729.

729 = 3 x 3 x 3 x 3 x 3 x 3

∛729 = ∛ (3 x 3 x 3 x 3 x 3 x 3)

Problem 5. Calculate the cube root of 216 by the prime factorisation method.

We have to find the cube root of 216.

Write the prime factorization of 216.

216 = 2 x 2 x 2 x 3 x 3 x 3

∛216 = ∛ (2 x 2 x 2 x 3 x 3 x 3)

= 2 x 3

Problem 6. Estimate the cube root of 2 by using the approximation method.

We have to estimate the cube root of 2.

Express 2 in the form of its prime factors. So we get,

2 = 1 x 1 x 2

Cube root of 2 = (1 + 1 + 2)/3

Problem 7. Estimate the cube root of 5 by using the approximation method.

We have to estimate the cube root of 5.

Express 5 in the form of its prime factors. So we get,

5 = 1 x 1 x 5

Cube root of 5 = (1 + 1 + 5)/3

The Cube Root of any number is another number which when multiplied by itself twice gives the number whose Cube Root is to be determined. Cube of root represented using the symbol ∛. The Cube Root of any number can also be represented in the form of an exponent as the number to the power ⅓. The Cube Root of any real number, say ‘k’, its Cube Root can either be written as ∛k or (k). We need to understand that the Cube and Cube Roots are inverse Mathematical operations. So, the Cube Root of a Cube of the number is the number itself.

### How to Find a Cube Root?

The Cube Root of any real number is obtained by either the prime factorization method or estimation method when the number whose square root is to be found is a perfect Cube number. However, in most cases, it is very much recommended to memorize the Cubes and Cube Roots of the first 25 natural numbers at the least. This will help the students and the facilitators to achieve excellent scores in competitive examinations. The table below gives the Cubes and Cube Roots of the first 20 natural numbers.

### Cubes and Cube Roots of the First 20 Natural Numbers

In this method, the number whose Cube Root is to be found is resolved completely into its prime factors. The identical prime factors are grouped such that three identical factors form one group. To determine the Cube Root, one factor from each group is collected and multiplied together.

The given number is completely resolved into its prime factors. It is always recommended to start the division with the lowest possible prime number and then go to the higher prime number when the quotient is not completely divisible by the number chosen.

The given number whose Cube Root is to be determined is 74088. The prime factorization can be summarized as shown in the figure below.

(Image will be Uploaded soon)

Write the number whose Cube Root is to be determined as the product of their primes.

The number 74088 can be written as the product of its primes as:

74088 = 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7

Divide the factors into groups consisting of three identical factors.

74088 as the product of its primes is rewritten as:

2 x 2 x 23 x 3 x 3 7 x 7 x 7

The Cube Root of the number is found as the product of one factor taken from each group in step 3.

So, the Cube Root of 74088 is found to be 2 x 3 x 7 = 42

### How to Find the Cube Root by Estimation Method?

The given number is divided into groups of 3 digits starting from the rightmost digit of the number. If any number is left out without forming a group of three, zeros are appended to its left to make it a group of 3 digits. However, we must take care that the place value of the digit is not altered by appending zeros.

Let us try finding out the Cube Root of 74088.

So, to find the Cube Root of 74088, we should divide the number into groups of three digits starting from the digit in the unit’s place.

From the first group starting from the right, note down the unit’s digit.

The first rightmost group in step 1 is 088 and the digit in its unit’s place is 8.

Estimate the digit in the unit’s place of the Cube Root of the given number using the lookup table given below.

The digit in the unit’s place obtained in step 1 is ‘8’ and hence the unit’s digit of the Cube Root of 74088 is also ‘2’.

Now, consider the second group from the right. Check the perfect Cube numbers between which this number lies. Suppose the number in this group lies between A and is closer to B, then the ten’s digit of its Cube Root is considered as B.

In the given number 74088, the second group of 3 digits from the right is 074.

This number lies between two perfect Cube numbers 64 and 125 i.e. 4. Because 74 is closer to 64 i.e. 4, the ten’s digit of the Cube Root of 74088 is 4.

The Cube Root of 74088 found using the estimation method is 42.

## FAQs on Cube and Cube Roots

3D Cube is solid and a physical object as cube numbers are the abstract concept which is a number we get after raising any number to it’s third power.

### What is the Cube of 3?

Cube of 3 is 27.

### What is the Cube of 4?

Cube of 4 is 64.

### How do you calculate the cube root of a number?

We can find the cube root of a perfect cube number using the prime factorization method or we can estimate the cube root using the steps explained in this same article.

### What is the Cube Root of 27?

A cube of 3 is 27, therefore cube root of 27 is 3.

### What is the Cube Root of 125?

A cube of 5 is 125, therefore cube root of 125 is 5.

### What is the relationship between Cubes and Cube roots?

Cube and Cube Roots are inverse operations that undo the effect of each other on any number. In other words, if we take any number and take it’s third power to find its cube and then find the cube root of the result. We will end up with the same number we have taken at first.

## How to Find Cube Root of a Number?

The cube root of a number can be determined by using the prime factorization method. In order to find the cube root of a number:

Example: Let us see how to find the cube root of 15625.

The cube root formula is the formula that is used to find the cube root of any substance. Suppose the cube of a number is x is y we can represent this as,

x3 = y

Now the cube root of y is calculated as,

y1/3 = x

The image added below represent the same.

## Applications of Cube Root Formula

Given below are a few major applications of cube root formula,

The cube root formula helps in the calculation of the cube root of any given number that is expressed in a radical form using the symbol ∛. It can be calculated by first finding out the prime factorization of the given number and then later applying the cube root formula. Suppose, x is any number such that, x = y × y × y.

The formula to calculate the cube root is given as:

where, y is the cube root of any number x. This also means that the number x would be a perfect cube if y has an integer value.

Given below are a few major applications of the cube root formula:

## Solved Example on Cube Root Calculator

Solved Example 1:

What is the cube root of 64?

The cube root of 64 is 4 because when 4 is multiplied by itself thrice, it gives 64

4 × 4 × 4 = 64.

Therefore, the cube root of 64 is 4 which is written as ∛64 = 4

Let us see how our cube root calculator gives the answer.

Enter the number 64 in the box.

The answer is given in the form of these steps:

Here, 64 is a perfect cube because it has an exact cube root.

There are certain numbers which are not perfect cubes because their cube root is in the decimal form. For example, the cube root of 34 is 3.24.

- -343
- 729

## Representation of Cube Numbers

## What Is a Cube Root Calculator?

A cube root calculator is a free online tool that calculates the cube root of a given number.

Cuemath’s cube root calculator helps you to find out the cube root of any number, in a few seconds, whether it is a perfect cube or not.

Note: Please enter upto six digits.

## Unit Digits in Cube Numbers

As we discussed in the properties of the cube

## Cube Root Formula for Negative Numbers

To find the cube root of any number, the prime factorization method is the best way out.

Let’s take a quick look at a couple of examples to understand the cube root formula, better.

Great learning in high school using simple cues

Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.

Book a Free Trial Class

## What is Cube?

Physically a cube is a solid figure which has all its sides equal. In mathematical terms, the cube of a number is the result of multiplying a whole number by itself twice, i.e., the whole number is used three times just like the sides of a cube. For example:

### Cube 1 to 20

The cube of natural numbers 1 to 20 is discussed in the table below,

### Properties of Cube of Numbers

Similar to positive integers we can find the cube of negative integers as well. It is the same for negative numbers as well. To find the cube of a negative number we just multiply the negative number three times by itself. As a product of two negative numbers is positive and the product of negative and positive numbers is negative. Thus, a cube of a negative number is always negative.

For example, a cube of -2 is nothing but (-2)×(-2)×(-2) = 4×(-2) = -8.

## Cube of Fractions

The cube root of a number can be calculated by first finding out the prime factorization of the given number and then later applying the cube root formula. Suppose, x is any number such that, x = y × y × y.

Cube root of x = ∛x = ∛(y × y × y) = y

This also means that the number x would a perfect cube if y has an integer value.

## List of Cube Root of Numbers

When we multiply a number three times by itself, the resultant number (product) is known as the cube of the original number. We call it a cube because it is used to represent the volume of a cube. In other words, a number raised to exponent 3 is called the cube of that number. For example, the cube of 3 is 27. That means 3 × 3 × 3 = 27, and it can be written as 33. Similarly, the cube of 4 is 64, and the cube of 5 is 125, and so on.

To find the cube of a number, first, multiply that number by itself, then multiply the product obtained with the original number again. Let us find the cube of 7 through the same process. We know that the cube of a number N is N × N × N. So, the cube of 7 is 7 × 7 × 7. Now, in order to find the cube of 7, we will first find the value of 7 × 7. This value is 49. Now, we will find 49 × 7. This is equal to 343. Hence, we can say that cube of the number 7 is 343.

### Cube of a Fraction

Similar to the cube of a number, the cube of a fraction can be found by multiplying it three times. For example, the cube of the fraction (2/5) can be written as 2/5 × 2/5 × 2/5. Simplifying it further, we get the value of the cube as (2 × 2 × 2) / (5 × 5 × 5). This is equal to (23/ 53) = 8/125.

The process to find the cube of a negative number is the same as that of a whole number and fraction. Here, always remember that the cube of a negative number is always negative, while the cube of a positive number is always positive. For example, let us try finding the cube of -7. We know that the cube of -7 is (-7) × (-7) × (-7). Now, in order to find the cube of (-7), we will first find the value of (-7) × (-7). This value is 49. Now, we will find 49 × (-7). This is equal to -343. Hence, we can say that cube of the number -7 is -343. The cube root formula for negative numbers is:

☛ Related Topics

Check these interesting articles related to the cube root.

## How to Use the Cube Root Calculator?

Example 1: Calculate the cube root of 343.

To find: Cube root of 343

Using the cube root formula,

Cube root of 343 = ∛343=∛(7×7×7)

Answer: The cube root of 343 = 7.

Example 2: Check whether 512 is a perfect cube or not.

To find: Whether 512 is the perfect cube or not.

Cube root of 512 = ∛512=∛(2×2×2×2×2×2×2×2×2)=∛(8×8×8)

= 8, which is an integer. Therefore, 512 is a perfect cube.

Answer: 512 is a perfect cube.

Example 3: Calculate Ron’s age if his age is the cube root of his grandmother’s age, while her present age is 64 years.

To find: Ron’s age if his age is the cube root of his grandmother’s age

His grandmother’s age = 64 years.

Ron’s age= cube root of 64 = ∛64 years = 4 years.

Answer: Ron’s age is 4 years

The cube root formula helps in the calculation of the cube root of any given number. The cube root of a number is defined as the number which when multiplied three times gives the original number. The cube root of a number is expressed in radical form using the symbol ∛. Let us understand the cube root formula using solved examples.

## FAQs on Cube Root

The cube root is the reverse of the cube of a number and is denoted by ∛. For example, ∛216, that is, the cube root of 216 = 6 because when 6 is multiplied thrice with itself, it gives 216. In other words, since 63 = 216, we have ∛216 = 6.

### What is the Difference Between Cube and Cube Root?

Cube root is used to solve cubic equations. They are also used to find the side length of a cube if its volume is given.

### How to Find the Cube Root of a Number?

Cube root can be simplified using the prime factorization method. First, do the prime factorization of the given number, then take out the common factors in groups of 3. Multiply these common factors taking only one from each group to get the answer.

### Is it Possible to Simplify the Negative Cube Root?

Yes, the simplification of negative cube roots is the same as positive cube roots. The only difference is the presence of a negative sign with the cube root of a negative number.

### What is Not a Perfect Cube?

A number is not a perfect cube if we cannot make 3 equal groups of factors of the number after doing the prime factorization. For example, 144 is not a perfect cube because there is no number that, when multiplied 3 times with itself, gives 144 as the product. In other words, if the cube root of a number is not an integer, then it is not a perfect cube.

### What is the Cube of an Odd Natural Number?

The cube of an odd natural number is always an odd number. For example, 53 = 125, 73 = 343, 93 = 729, etc.

### Can the Cube Root of Any Odd Number be Even?

No, the cube root of an odd number is always odd. For example, the cube root of 27 = (27)1/3 = 3. Here, both 3 and 27 are odd numbers.

### How to Calculate Cube Root of Any Number?

The easiest and basic method to find the cube root of any number is the prime factorization method.

### What is the Cube Root Formula in Algebra?

In math, the cube root formula is used to represent any number in the form of its cube root, such as for any number x, its cube root will be 3√x = x1/3. For example, the cube root of 125 is 5 1251/3 = (53)1/3 = 5.

### What is the Cube Root Formula for Negative Numbers?

Yes, it is possible to find the cube root of negative numbers. For example, -64 = (-4) × (-4) × (-4). We can write -64 as the product of 3 negative 4’s. Thus, ∛-64 = -4 because the product of three negative values gives us a negative result.

### How to Use Cube Root Formula?

Now, let’s learn about them in detail.

### By Prime Factorization

Prime factorization is a method through which you can easily determine whether a particular figure represents a perfect cube. If each prime factor can be clubbed together in groups of three, then the number is a perfect cube.

For example: Let us consider the number 1728.

As prime factorization of 1728 is 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be rewritten as 23 × 23 × 33.

Here, all the numbers can be clubbed into groups of three. So we can definitely say that 1728 is a perfect cube. In fact, the cube root of 1728 is a product of numbers taken one from each group i.e., 2 × 2 × 3 = 12.

Example 1: Find the cube root of 216.

216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 * 33 = 63

Hence, the cube root of 216 is 6.

Example 2: Find the cube root of 5832.

5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

5832 = 23 × 33 × 33 = 183

Hence, the cube root of 5832 is 18.

### Cube Root of a Cube Number using Estimation

Step 1: Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number.

So 117649 has two groups, and the first group is (649) and the second group is (117).

Step 2: The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since 649 ends with 9, the cube root’s unit digit is 9.

Step 3: Find the cube of numbers between which the second group lies. The other group is 117.

We know that 403= 64000 i.e., second group for cube of 40 is 64, and 503= 125000 i.e., second group for 50 is 125. As 64 < 117 < 125. Thus, the ten’s digit of the requred number is either 4 or 5 and 50 is the least number with 5 as ten’s digit. Thus, 4 is the ten’s digit of the given number.

So, 49 is the cube root of 117649.

Example: Estimate the Cube root of the number 357911.

Let’s take another cube number, say 175616.

Step 1: Starting from the rightmost digit, group the digits in threes. So, the first group is (616) and the second group is (175).

Step 2: The unit digit of the first group (616) is 6, which corresponds to the unit digit of the cube root 6.

Step 3: Find the cubes of numbers between which the second group lies. We know that 43³ = 79507 and 44³ = 85184. Since 175 is between 79507 and 85184, the tens digit of the required number is 4.

Therefore, the cube root of 175616 is 46.

### Hardy-Ramanujan Numbers

Numbers like 1729, 4104,13832, etc. are known as Hardy – Ramanujan Numbers because they can be expressed as the sum of two cubes in two different ways. A number n is said to be Hardy-Ramanujan Number if

n = a3 + b3 = c3 + d3

The first four Hardy-Ramanujan numbers are:

91 = 63 + (-5)3 = 43 + 33

Note: The number 1729 is also sometimes referred to as the taxicab number as it was the number of the taxi taken by Dr. Hardy while going to meet Ramanujan at the hospital.

## What is Cube Root?

The cube root of a number is a value that, when multiplied three times, gives that number. For example, 3 × 3 × 3 = 27, so the cube root of 27 is 3.

### Symbol of Cube Root

The cube root of a number is a number which, when multiplied by itself three times, gives the original number.

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## Perfect Cubes

A perfect cube is a positive integer equal to some other positive integer raised to the third power. In other words, when we multiply an integer by itself the resulting number is called a perfect cube. For example, 125 is a perfect cube since 125 = 5 × 5 × 5 = 53. However, 121 is not a perfect cube because there is no integer n such that n3 = 121 = 112. Some other examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, . . . , etc.

## FAQs on Cube Root Formula

In math, the cube root formula is used to represent any number in the form of its cube root, such as for any number x, its cube root will be 3√x = x1/3. For example, the cube root of 125 is 5 because 5 × 5 × 5 = 125.

Yes, it is possible to find the cube root of negative numbers. For example, -64 = (-4) × (-4) × (-4). We can write -64 as the product of 3 negative 4’s. Thus, ∛-64 = -4 because the product of three negative values gives us a negative result.

In order to use the cube root formula

### How To Write Cube Root Formula in Words?

The cube root of any number is the number raised to the power of 1/3.

## Cube Root Definition

When we think about the words cube and root, the first picture that might come to our mind is a literal cube and the roots of a tree. Isn’t it? Well, the idea is similar. Root means the primary source or origin. So, we just need to think ‘cube of which number should be taken to get the given number’. In mathematics, the definition of cube root says that Cube root is the number that needs to be multiplied three times to get the original number. Now, let us look at the cube root formula, where y is the cube root of x. ∛x = y. The radical sign ∛ is used as a cube root symbol for any number with a small 3 written on the top left of the sign. Another way to denote cube root is to write 1/3 as the exponent of a number.

Cube root is an inverse operation of the cube of a number.