In mathematics, a cube root of a number is a number such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of , denoted , is , because 23 = 8, while the other cube roots of are and . The three cube roots of are

In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always the cube of a nonzero number has more than one complex cube root and its principal cube root may not be the number that was cubed. For example, , but

The cube roots of a number x are the numbers y which satisfy the equation

For any real number x, there is one real number y such that y3 = x. The cube function is increasing, so does not give the same result for two different inputs, and it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number.

The three cube roots of 1

If x and y are allowed to be complex, then there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are:

The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.

Plot of the complex cube root together with its two additional leaves. The first image shows the main branch, which is described in the text.

Riemann surface of the cube root. One can see how all three leaves fit together.

For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula

If we write x as

where r is a non-negative real number and θ lies in the range

then the principal complex cube root is

This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance will not be −2, but rather 1 + i.

This difficulty can also be solved by considering the cube root as a multivalued function: if we write the original complex number x in three equivalent forms, namely

The principal complex cube roots of these three forms are then respectively

Unless , these three complex numbers are distinct, even though the three representations of x were equivalent. For example, may then be calculated to be −2, 1 + i, or 1 − i.

- Impossibility of compass-and-straightedge construction
- Appearance in solutions of third and fourth degree equations
- The Cube Root Symbol
- Find the Cube of Fraction
- You Can Also Cube Negative Numbers
- Cube Roots (For Integer Results 1 through 10)
- A Few Properties of Cube Root
- What is a Cube Root Calculator?
- How can you use a Cube Root Calculator?
- 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?
- What is a Cube Root?
- Symbol of the Cube Root
- Cube Root of Negative Numbers
- Properties of Cube Roots
- What is Cube?
- Cube 1 to 20
- Properties of Cube of Numbers
- Cube of Negative Numbers
- Cube of Fractions
- What is Cube Root?
- Symbol of Cube Root
- How to find the Cube Root of a Number?
- By Prime Factorization
- Cube Root of a Cube Number using Estimation
- Hardy-Ramanujan Numbers
- 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?
- Prime Factorization Method
- Approximation Method
- Sample Problems
- Answers to Questions (FAQ)
- How to calculate a cube root on a calculator?
- How to compute a cube root on Excel?
- How to simplify a cube root?
- What is a cubic number?
- What is the cube root of unity (1)?
- How to write a cube root?
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## Impossibility of compass-and-straightedge construction

Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837 Pierre Wantzel proved that neither of these can be done with a compass-and-straightedge construction.

The method is simply averaging three factors chosen such that

at each iteration.

Halley’s method improves upon this with an algorithm that converges more quickly with each iteration, albeit with more work per iteration:

This converges cubically, so two iterations do as much work as three iterations of Newton’s method. Each iteration of Newton’s method costs two multiplications, one addition and one division, assuming that is precomputed, so three iterations plus the precomputation require seven multiplications, three additions, and three divisions.

Also useful is this generalized continued fraction, based on the nth root method:

If x is a good first approximation to the cube root of a and y = a − x3, then:

The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence.

## Appearance in solutions of third and fourth degree equations

Cubic equations, which are polynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a rational number). If two of the solutions are complex numbers, then all three solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number.

Quartic equations can also be solved in terms of cube roots and square 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

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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Finding the cube root by multiplying its nesting fourth roots looks okay but assert fails.

Whenever you come across the word cube root the two words that come into our mind are cube and roots of a tree. Actually the concept is a bit similar in this sense, root actually refers to the primary source of origin. So we should think about what number you should refer to. That cube will give you a particular number that you are looking for.

Cube root of a number x is a number y only when y×y×y= x. All the nonzero real numbers have one real cube root and along with it a pair of complex conjugate cube roots, and all non zero complex numbers have three different complex roots that are cube roots.

The cube root of any number in brief can be defined as the factor that we multiply by itself three times to get the particular numbers. Remember that the cube root of a number is exactly opposite of the cubing of a number.

The process of cubing is similar to squaring, only that the number is multiplied three times instead of two times as in squaring. The exponent used for cubes is 3, which is also denoted by the superscript³. Examples are 4³ = 4*4*4 = 64 or 8³ = 8*8*8 = 512 etc.

To find the volume of the cube, we have volume = side3, but if we want to find the side of a cube we have to take the cube root of the volume. Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbol is 3√.

Let’s suppose we need to find the value of cube root of 2 is a value that is obtained by multiplying that number three times. It is expressed in the form of ‘3√2’. The meaning of cube root is basically the root of a number that is generated by taking the cube of another number. Hence, if the value of 3√2=x, then x =2 and we need to find the value of x.

(image will be uploaded soon)

We can define the cube root of a number as a special value that, when used in a multiplication exactly three times, gives us that number.For example, 3 × 3 × 3 equals 27, so the cube root of 27 is 3.

### The Cube Root Symbol

The special symbol given below signifies the «cube root», it is known to be the «radical» symbol (the symbol can be used for square roots) and with a little three to mean cube root.

You can use it like this, the cube root of 27 is : 3√27=3 (we say «the cube root of 27 equals to 3»)

### Find the Cube of Fraction

You can find the cube of a fraction the same way as you are finding the cube of a number. You just need to multiply the fraction three times. Just take the example of ⅔. You can get the cube of this fractional number by first multiplying the number in the numerator 3 times that is here you can first multiply the number 2 three times, 2×2×2 and as a result you will get 8 that is the Cube of the number 2 which is in your numerator.

Now move towards the denominator, here in your denominator there is 3, you can get its cube by multiplying it 3 times, that is 3×3×3 hereafter cubing the number 3 you are going to get 27 as a result, which is the Cube of your number 3, registered here as the denominator. So as a result you are going to get 8/27. This 8/27 is the Cube of your number ⅔ which you can get either by doing the multiplication of these numbers separately or you can do the multiplicity of the whole fraction three times, that is ⅔ × ⅔ × ⅔. Giving you the result of 8/27.

### You Can Also Cube Negative Numbers

Have a look at this:When we cube +5 we generally get +125: +5 × +5 × +5 = +125

When we cube −5 we get the number −125: −5 × −5 × −5 = −125. So the cube root of the number −125 is equal to −5

### Cube Roots (For Integer Results 1 through 10)

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself.

Let’s see for example,

23 =8, or the cube root of the number 8 is 2

33 = 27, or the cube root of the number 27 is 3

43 = 64, or the cube root of 64 is 4

53 = 125, or the cube root of 125 is 5

The symbol of the cube root is a

Thus, the cube root of 125 is represented as 3√125=5 and that of 27 can be represented as 3√27 equals 3 and so on.

We know that the cube of any number is found by multiplying that number three times. And the cube root of a number can be defined as the inverse operation of cubing a number.

If the cube of a number 63 = 216

Then the cube root of ∛216 is equal to 6.

Cube root of any largest number can be easily found in four ways:

### A Few Properties of Cube Root

A perfect cube of an integer is that integer which is actually equal to some other integer raised to the third power. We refer to raising the number to the third power as cubing the number.

Perfect Cube :1 8 27 64 125 216 343 512 729 1000 .

### What is a Cube Root Calculator?

Cube root calculator is a tool that will help you to find the cube root of a particular number. This calculator is free. By using the cube root calculator you can find the cube root of a number free without giving any charge.

### How can you use a Cube Root Calculator?

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.

Now we are getting closer, but slowly at this point, we can use a calculator and it says:

What is the Cube Root of 1728?

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.

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.

To know the cube root formula, you need to know the cube formula.

Cube of any digit, forms by multiplying the digit by itself three times. For instance, to find the cube:

We need to multiply 5 three times in the case of : 5 × 5 × 5 = 125

Note: We need to write down “5 cube” as (the little 3 on the top means the number appears three times during the multiplication process.)

The Cube Formula for any value ‘x’ can be given as,

If we group the prime factors of a number in triples of equal factors, then that number is known as a perfect cube. In order to check whether a number is a perfect cube or not, we need to find its prime factors and then group together triplets of the prime factors. If no factor is left out after the process then the number is known to be a perfect cube.

Here are a few more examples of perfect cube numbers:

### What is a Cube Root?

A cube root goes the opposite direction. For example, 3 is cubed to give the result 27 so the cube root of 27 will be 3.

Therefore, the cube root of a number is a special number which when cubed gives the original number as a result. The cube root of 27 is 3 because 3 is cubed to produce 27.

### Symbol of the Cube Root

Check the example below for detail:

5 Cube =

Thus, the cube root of 125 is 5. The number 125 is a perfect cube.

### Cube Root of Negative Numbers

The cube of a negative number will also be a negative number.

### Properties of Cube Roots

The table given below has the cubes of all the number between 11 to 20

A number having 1 in its unit digit, will also have 1 in the unit digit of their cubes.

1³ = 1

11³ = 1331

21³ = 9261

31³ = 29791

The cubes of 1, 4, 5, 6, 9, and 0 also have the same digits in its unit digits.

14³ = 2744

15³ = 3375

16³ = 4096

20³ = 8000

The cube of numbers ending 2 as unit digit will have 8 in its unit digit. Similarly, the cube of the numbers ending in unit digit 8 will have a unit digit 2.

12³ = 1728

18³ = 5832

The cube of the numbers with 3 as unit digit will have a unit digit 7. Similarly, the cube of numbers with the unit digit 7 will have a unit digit 3.

13³ = 2197

27³ = 19683

The cubes of all even numbers are even. Also, the cubes of odd numbers are all odd.

18³ = 5832 (even)

27³ = 19683 (odd)

The sum of the cubes of the first natural numbers m is equal to the square of their sum.

1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)²

1 + 8 + 27 + 64 = (10)²

100 = 100

Cube root of any large number can be easily found in four ways:

Let’s know how to find the cube root of any number

Question 1)What is the cube root of 1728?

The factors of 1728 are given as,

Solution: Cube root of 27 is 3. (3 × 3 × 3)

Cube root of 125 is 5 (5 × 5 × 5)

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.

## 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

## Cube of Negative 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

As we discussed in the properties of the cube

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.

## 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 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.

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

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.

## 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.

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

Tool to compute a cube root. The cube root for a number N, is the number that, multiplied by itself than again by itself, equals N.

## Answers to Questions (FAQ)

dCode software allows positive of negative numbers (complex roots) and answers an exact value or an approximate one (the precision can be adjusted by defining the precision: a minimum number of significant digits)

### How to calculate a cube root on a calculator?

Example: Calculate the cube root of 64 on a calculator by typing 64 ^ (1/3) = (answer is 4)

### How to compute a cube root on Excel?

On a spreadsheet like Microsoft Excel, use the same formula as for a calculator, for a value in A1 write A1^(1/3) or POWER(A1;1/3)

### How to simplify a cube root?

The root simplifier will attempt to factor the expression under the root with a perfect cube.

### What is a cubic number?

A cubic number is the cube of an integer (cubed value).

Example: $ 2 $ is an integer, $ 2^3 = 2 imes 2 imes 2 = 8 $ then $ 8 $ is a square number.

If the cube root of a number $ x $ is an integer (relative, without decimal part), then $ x $ is a cubic number.

The first perfect cubes are:

### What is the cube root of unity (1)?

In some software, cbrt stands for cube root abbreviation cb of cube and rt for root, similar to sqrt for square root.

Example: cbrt(8)=2

### How to write a cube root?

The Unicode standard proposes the symbol U+221B ∛

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