Question

Find the cubic root of following number 10648.

Answer 1

Hint: The cube root of any number can be obtained by using factorization method. A number can be represented by multiplication of all its factors. Factors of a number can be obtained by evenly dividing the number by a series of prime factors until we obtain the quotient of 1 and remainder of 0. Series of prime factors is 2,3,5,7,11,13,17,19,23,29…. So start dividing the given number by prime numbers until we obtain a quotient of 1 and remainder of 0. After that take the cube root of multiplication of all factors.

Complete step-by-step answer:
Here the given number is 10648. We have to find the cube root of 10648.
To find the cube root of 10648 we will use the factorization method.
In the factorization method we will get the factors of number. Factors of a number can be obtained by evenly dividing the number by a series of prime factors until we obtain the quotient of 1 and remainder of 0. Series of prime factors is 2,3,5,7,11,13,17,19,23,29…. So start dividing the given number by prime numbers until we obtain a quotient of 1 and remainder of 0.
Let's start dividing the number 10648 by the first prime number 2.
$\dfrac{{10648}}{2} = 5324$. Here the reminder is 0. So 2 is the first factor of 10648.
Now, $\dfrac{{5324}}{2} = 2662$. Here the reminder is 0. So 2 is the second factor of 10648.
Now, $\dfrac{{2662}}{2} = 1331$. Here the reminder is 0. So 2 is the third factor of 10648.
Now, $\dfrac{{1331}}{2} = 665.5$. Here 1331 is not evenly divided by 2. There is a reminder of 1. So 2 will not be the factor now.
Now trying with next prime number 3,
So, $\dfrac{{1331}}{3} = 443.67$. Here 1331 is not evenly divided by 3. There is a reminder of 2. So 3 will not be the factor.
Now trying with next prime number 5,
So, $\dfrac{{1331}}{5} = 266.2$. Here 1331 is not evenly divided by 5. There is a reminder of 1. So 5 will not be the factor.
Now trying with next prime number 7,
So, $\dfrac{{1331}}{7} = 190.14$. Here 1331 is not evenly divided by 7. There is a reminder of 1. So 5 will not be the factor.
Now trying with next prime number 11,
So. $\dfrac{{1331}}{{11}} = 121$. Here the reminder is 0. So 11 is the fourth factor of 10648.
Now, $\dfrac{{121}}{{11}} = 11$. Here the reminder is 0. So 11 is the fifth factor of 10648.
Now, $\dfrac{{11}}{{11}} = 1$. Here reminder is 0 and quotient is 1. So, 11 is the sixth factor of 10648.
So number 10648 can be represented by multiplying all prime factors of numbers.
So, $10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11$
Converting into exponential form, $10648 = {2^3} \times {11^3}$
Taking cube root on both side,
$\sqrt[3]{{10648}} = \sqrt[3]{{{2^3} \times {{11}^3}}}$
Simplifying, $\sqrt[3]{{10648}} = 2 \times 11$
So, $\sqrt[3]{{10648}} = 22$.

So, the cube root of number 10648 is 22.

Note: There is an alternative method for finding the cube root of a number which is called estimation method. In this method start making a group of three digits. Start making a group from a unit place.
Here for number 10648, first group is 648 and second group is 10.
First group will give the unit digit of the cube root. Here first group 648 is ending with 8. Cube root of 8 is 2. So, the unit digit of the cube root of our required number will be 2.
Second group will give tens of places for the cube root. Now the second group is 10. As we know that $8\langle 10\langle 27$ means ${2^3}\langle 10\langle {3^3}$. So consider the smallest number 2, it will become tens place for the required cube root. So, $\sqrt[3]{{10648}} = 22$.