Can we find cube root by prime factorization method?

The prime factorization method can be used to find the cube root of an integer that is a perfect cube.

Step 1: Find the number whose cube root you're looking for.

Step 2: Divide the number by the smallest prime number feasible until it is no longer divisible by that prime number. If the number isn't divisible by the lowest prime number assumed, move on to the next higher prime number and repeat the process.

Step 3: As the product of their primes, express the integer whose square root is to be computed.

Step 4: Each set of three identical components is formed.

Step 5: Locate the product by utilising one element from each group. The resulting product is equal to the number's cube root.

\begin{array}{l} =\sqrt[3]{(}(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(2 \times 2 \times 2)) \\ =\sqrt[3]{\left(2^{3} \times 2^{3} \times 2^{3}\right)} \\ =2 \times 2 \times 2 \\ =8 \end{array}

Cube root of a number can be found by a very simple method which is the prime factorization method. Cube root is denoted by ‘∛ ‘ symbol. Example: ∛8 = ∛(2 × 2 × 2) = 2. Since, 8 is a perfect cube number, it is easy to find the cube root of a number.

Finding the cubic root of non-perfect cube number is a little complex process but can be mastered easily. To find the cube root of any number, we need to find a number which when multiplied three times by itself gives the original number.

Let us learn here finding cube root using prime factorisation method and solved examples. Also, find the cubes and cube roots of 1 to 15 numbers here in the table given.

Table of contents:

Definition

The cube root of any number say ‘a’, is the number ‘b’, which satisfies the equation given below:

b3 = a

This can be represented as:

a = ∛b

How to Find Cube Root of a Number

Cube root is the inverse process of calculating the cube of a number. It is denoted by the symbol ‘∛’. Let us see some examples here now.

To find the cube root of a number 27, we want a number which when multiplied thrice with itself shall give 27. We can write,

27 = 3 × 3× 3 = 33

Taking cubic root on both the sides;

or ∛27 = ∛33

Therefore, the cube-root of 27 is 3.

Please note that we will only consider the positive values cube roots of the natural numbers.

Cube Root of 2

Let us consider another example of number 2. Since 2 is not a perfect cube number. It is not easy to find the cube root of 2. With the help of the long division method, it is possible to find the cube roots for non-perfect cube numbers. The approximate value of the ∛2 is 1.260. 

We can estimate the ∛2 by using the trick here.

Since, 2 = 1 x 1 x 2

Cube root of 2 is approximately equal to (1 + 1+2)/3 = 4/3 = 1.333..

Cube root of 4

Again 4 is a number, which is not a perfect cube. If we factorise it, we get:
4 = 2 x 2 x 1
Hence, we can see, we cannot find the cube root by simple factorisation here.
Again, if we use the shortcut method, we get:
∛4 is equal to (2+2+1)/3 = 1.67
The actual value of ∛4 is 1.587, which is approximately equal to 1.67.

Cubes and Cube Roots List of 1 to 15

NumberCube(a3)Cube root ∛a111.000281.2603271.4424641.58751251.71062161.81773431.91385122.00097292.0801010002.1541113312.2241217282.2891321972.3511427442.4101533752.466

Also, read:

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

Cube Root By Prime Factorisation Method

We can find the cube-root of a number by the method of prime factorisation. Consider the following example for a clear understanding:

2744= 2 × 2× 2 × 7 ×7 × 7= (2 × 7 )3

Therefore, the cube root of 2744 = ∛2744 = 2 × 7 = 14

Finding Cube Root Using Division Method

For finding the cube root using the division method is similar to using the long division method or manual square method. Make a pair of 3 digit numbers from the back to front. Next step is to find the number whose cube root is less than or equal to the given number. Now, subtract the obtained number from the given number and write down in the second number. After this step, it is necessary to find the multiplication factor for the further process in the long division method, which comes by multiplying the first number obtained. Similarly continue the above process, to find the cube root of a number. This long division process is used when the given number is not a perfect cube number. Finding the cube root of a number using this process takes a long time.

Cube Root of 64

Since 64 is a perfect cube of 4, therefore, it is easy to find its cube-root by the prime factorisation method.
64 = 2 x 2 x 2 x 2 x 2 x 2
∛64 = ∛(2 x 2 x 2 x 2 x 2 x 2)
= 2 x 2
= 4

Cube Root of 216

Since, 216 is perfect cube of 6, hence we can find the cube root of 216 by factorisation.
216 = 2 x 2 x 2 x 3 x 3 x 3
∛216 = ∛(2 x 2 x 2 x 3 x 3 x 3)
∛216 = 2 x 3
∛216 = 6

Cube Root of 343

Let us find the cube root of 343 with the help of the the prime factorisation method.

Dividing 343 by smallest prime factor, till we get the remainder as 1. Follow the below steps;

Cube root of 343

Therefore, 343 = 7 × 7 × 7

And, ∛343 = 7

Cube Root of 512

To find the cube root of 512 we have to factorise it first.
The prime factorisation of 512 can be written as:
512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Taking the cube roots both the sides, we get;
∛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

Cube Root of 729

Now, let’s find the cubic root of 729.

Cube root of 729

729 = 3 × 3 × 3 × 3 × 3 × 3 = 9 × 9 × 9

Therefore, the cube root of 729 i.e. ∛729 = 9

Cube Root Questions

Question 1: Solve: ∛24389

Solution:

Prime factors = 29×29×29 = 293
Therefore, ∛24389= 29.

Question 2: Find ∛46656 by the method of prime factorization.

Solution:

Let us first find the prime factors:

Cube Root of Numbers

∛46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 = 23 × 23 × 33 × 33 = (2 × 2 × 3 × 3)3

Therefore, ∛46656= 36.

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