# How to Find Cube Root? Техника

### Cube Root Calculator

Here is the answer to questions like: Cube root of 216 or what is the cube root of 216?

Derivative of. In this article, we will find the derivative of the cube root of by applying the power rule of derivatives. We will also use the limit definition to evaluate the derivative of the cube root of .

The cube root of is a number when multiplied by itself two times will be . If a cube has a volume of 125 unit3, then we use the value of the cube root of to find the length of the cube.

The value of the cube root of is .

Let us now calculate the cube root of

#### What is the Cube Root of 125?

We know that 125=5×5×5. As this is a product of three number of ‘s. we can write

Taking cube root on both sides, we have

As cube root can be written as power , we get that

= 53×1/3   as we know that (am)n = am×n

= 51 = 5

So the cube root of is .

#### Is 125 a perfect cube number?

Note that the value of the cube root of is . To prove this by the prime factorization method, we will at first factorize

As has the unit digit , it will be divisible by . So we have

By the above logic, we write

So finally we get

Note that the above is the prime factorization of .

Taking cube root on both sides of , we get that

∴  the cube root of is .

#### Is Cube Root of 125 Rational?

By definition, we know that a rational number can be expressed as where and q are integers with .

So 5 can be written as p/q with .

is a rational number.

⇒ the cube root of is also a rational number.

Conclusion: The cube root of 125 is not an irrational number.

The cube root of a number is denoted by

The cube root of a number x is that number whose cube gives x. We denote the cube root of x by ∛x

Thus, 3√64 = cube root of 64 = 3∛4 × 4 × 4 = ∛4³ = 4

(i) Since (2 × 2 × 2) = 8, we have ∛8 = 2

(ii) Since (5 × 5 × 5) = 125, we have ∛125 = 5

### Method of finding the cube root of a given number by factorization

Express the given number as the product of primes.

Make groups in triplets of the same prime.

Find the product of primes, choosing one from each triplet.

This product is the required cube root of the given number.

If the group in triplets of the same prime factors cannot complete, then the exact cube root cannot be found.

### Solved Examples of Cube Root using step by step with explanation

1. Evaluate the cube root: ∛216

By prime factorization, we have 216 = 2 × 2 × 2 × 3 × 3 × 3

= (2 × 2 × 2) × (3 × 3 × 3)

Therefore, ∛216 = (2 × 3) = 6

2. Evaluate the cube root: ∛343 343 = 7 × 7 × 7

= (7 × 7 × 7).

Therefore, ∛343 = 7

3. Evaluate the cube root: ∛2744 2744 = 2 × 2 × 2 × 7 × 7 × 7

= (2 × 2 × 2) × (7 × 7 × 7).

Therefore, ∛2744 = (2 × 7) = 14

### Cube Root of a Negative Perfect Cube

Let (a) be a positive integer. Then, (-a) is a negative integer.

We know that (-a)³ = -a³.

Therefore, ∛-a³ = -a.

Thus, cube root of (-a³) = -(cube root of a³).

Thus, = ∛-x = — ∛x

Find the cube root of (-1000).

We know that ∛-1000 = -∛1000

Resolving 1000 into prime factors, we get 1000 = 2 × 2 × 2 × 5 × 5 × 5

= (2 × 2 × 2) × (5 × 5 × 5)

Therefore, ∛1000 = (2 × 5) = 10

Therefore, ∛-1000 = -(∛1000) = -10

### Cube Root of Product of Integers

We have, ∛ab = (∛a × ∛b).

1. Evaluate: ∛(125 × 64).

(∛125 × 64)

= ∛125 × ∛64

= (5 × 4)

2. Evaluate: ∛(27 × 64).

(∛27 × 64)

= ∛27 × ∛64

= (3 × 4)

= ∛216 × ∛-343

### Cube Root of a Rational Number

We define: ∛(a/b) = (∛a)/(∛b)

### Cube Root of Fractions

Cube root of a fraction is a fraction obtained by taking the cube roots of the numerator and the denominator separately.

If a and b are two natural numbers, then ∛(a/b) = (∛a)/(∛b)

### Cube Root of Decimals

Express the given decimal in the fraction form and then find the cube root of the numerator and denominator separately and convert the same into decimal.

Find the cube root of 5.832.

Converting 5.832 into fraction, we get 5832/1000

Now ∛5832/1000 = ∛5832/∛1000

= ∛(2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3)/∛(2 × 2 × 2 × 5 × 5 × 5)

= 2 × 3 × 3/2 × 5

Cube and Cube Roots

To Find if the Given Number is a Perfect Cube

Method for Finding the Cube of a Two-Digit Number

Table of Cube Roots

Cube and Cube Roots — Worksheets

Worksheet on Cube

Worksheet on Cube and Cube Root

Worksheet on Cube Root

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## Definition – What is a cubed root?

The cubed root of a number is the number that, when multiplied by itself three times (number x number x number), gives the original number.

For example, the cubed root of 27 is 3, as 3 x 3 x 3 = 27.

The cubed root of 125 is 5, as 5 x 5 x 5 = 125.

Unlike the square root, the cubed root is always positive.

The opposite of the cubed root is a cubed (power of 3) calculation.

In geometry cubed root can be used to find the length of a side of a cubed when the volume is known.

## Formula – How to calculate the cubed root of a number

There is no quick formula to calculate a cubed root. Most calculators use some form of trial and error.

Most calculators use trial and error to find a cubed root. Trial and error works well for perfect cubes. It can be very time consuming for non-perfect cubes because there are many decimal places.

To find cubed root by trial and error:

Example – Find the cubed root of 512 using trial and error:

### Method 2 – Quickly find roots from perfect cube numbers

This method makes it quicker to find the root of a perfect cube number. If the number is not a perfect cube root number, however, this method won’t give an answer.

## How do you type cubed root?

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

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:

#### 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

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,

### What are Perfect Cubes?

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 Roots (For Integer Results 1 Through 10)

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)

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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)

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.

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.

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.

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### 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

### What is the Meaning of Cube Root?

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:

## Perfect Cube Roots Table 1-100 ### Disclaimer

While every effort is made to ensure the accuracy of the information provided on this website, neither this website nor its authors are responsible for any errors or omissions. Therefore, the contents of this site are not suitable for any use involving risk to health, finances or property.

## What is the derivative of cube root of x?

First Method of Finding Derivative of Cube Root of x: At first, we will calculate the derivative of cube root x by the power rule of derivatives. See the below steps.

Therefore, we get that

Finally, simplify the above expression. Thus from we have

Differentiating both sides with respect to \$x\$, we obtain that

Also Read: Derivative of Square Root of x

Now, we will find the derivative of cube root of \$x\$ by the first principle.

## What is cube root?

A cube root of a number a is a number x such that x3 = a, in other words, a number x whose cube is a. For example, 6 is the cube root of 216 because 63 = 6•6•6 = 216, -6 is cube root of -216 because (-6)3 = (-6)•(-6)•(-6) = -216.

## FAQs on Cube Root of 125

Q1: Find the cube root of 125.

Answer: As 125 = 5×5×5, the cube root of 125 is 5.

Q2: Is 125 a perfect cube number?

Answer: Yes, 125 is a perfect cube number as 125=53.

## Application of Derivative of cube root of x

From above we get the derivative of root \$x\$ which is given below.

This will help us to find the derivatives of many functions involving cube roots by the chain rule of derivatives. At first, we will calculate the derivative of the cube root of .

Next, we will find the derivative of the cube root of , that is, the derivative of

Q1: What is the derivative of cube root of x?

Ans: The derivative of cube root of x is 1/3×2/3.

Q2: How a cube root of x defined?

A cube root of a number a is a number x such that x3 = a, in other words, a number x whose cube is a. For example, 0 is the cube root of 0 because 03 = 0•0•0 = 0, -0 is cube root of -0 because (-0)3 = (-0)•(-0)•(-0) = -0.

## Derivative of Cube Root x by Logarithmic Differentiation

Now, we will find the derivative of the cube root of x with the help of the logarithmic derivative. Note that cube root x can be written as . Write

We will take natural logarithms on both sides (i.e, logarithm with base e, denoted by ln). Thus we get that

ln y = 1/3 ln x

Differentiating with respect to x, we have

## Derivative of cube root of x from first principle

Myltiplying the numerator and the denominator of by the above quantity, we get that

Thus, the derivative of the cube root of by first principle is

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