Cube root of a number gives a value, which results in the original number when multiplied by itself thrice. It is denoted by the symbol ‘3√’. To find the cube root of a number, easily, we can use the prime factorisation method. But this method is applicable only for perfect cubes. Let us see some examples here to evaluate the cube root.
Prime factorisation of 27 =3x3x3
Hence, 3√(3x3x3) = 3√33 = 3
We can write 1000 as;
1000 = 2x2x2x5x5x5 = (2x2x2)x(5x5x5) = 23 x 53 = (10)3
Hence, 3√1000 = 3√(10)3 = 10
We can write, 216 as;
216 = 2x2x2x3x3x3 = 23x33 = 63
3√216 = 3√63 = 6
From the above examples, we can see that by finding the prime factors of the numbers under the cubic root, we can simply get the original numbers.
Now let us see how to find the cube root of imperfect cubes and large cubes.
How to Find Cube Root of Large Perfect Cubes
We can easily find the cubic root of a large natural number by using the estimation method. To use this method, we firstly need to memorise the cube table for numbers 1 to 10, to do fast calculation.
With the help of the table above we will be using the estimation method. Let us understand with the help of examples.
1. Cube root of 2197 is equal to?
Solution: We can also find the value of 3√2197 using the prime factorisation method. But it could take us more time to factorise a large number. Hence, we will use the estimation method here.
First, we will take the digit at the unit’s place.
At the unit place, we have 7.
Now consider the cube table, the cube of which number has 7 as at the unit digit place.
We see, 33 = 27.
That means the cube root of 2197 will have 3 at the unit place.
Now ignore the last 3 digits of 2197, i.e. 197
Taking 2 as a reference number, we can see from the cubes table 2 lies between 1 and 8.
Here, we choose the lower number here.
Therefore, we get the two-digit of the answer.
Hence, 3√2197 = 13
2. Find the cube root of 857375.
Solution: We will divide the number 857375 into two groups, i.e., 857 & 375.
Let 375 is the first cube and 857 is the second cube.
375 will give a unit place digit of the required cube root and 857 will give the second digit.
Now 375 has 5 at the unit place. We know the cube of 5 will give the cube root as 5. So at the unit place, we get 5 for the required cube root.
Now we take another group which is 857. So, from the cubes table, we observe the value 857 lies between cube of 9 and 10, i.e., 729 < 857 < 1000. So, we take here a lower number which is 9.
Hence, 3√857375 = 95
Cube Root of Non-Perfect Cubes
There are many numbers which are not perfect cubes and we cannot find the cube root of such numbers using the prime factorisation and estimation method. Hence, we will use here some special tricks to find the cube root.
Let us find the cube root of 150 here. Clearly, 150 is not a perfect cube.
Now we would see 150 lies between 125 (cube of 5) and 216 (cube of 6). So, we will consider the lower number here, i.e. 5.
Divide 150 by square of 5, i.e., 150/25 = 6.
Now subtract 6 from 5 (whichever is greater) and divide it by 3. So,
6-5 = 1 & ⅓ = 0.333
At the final step, we have to add the lower number which we got at the first step and the decimal number obtained.
So, 5+0.33 = 5.33
Therefore, the cube root of 150 is 3√150 = 5.3
If we find the value in the calculator, this value is approximately equal to the actual value, i.e. 5.314.