Question

Find the cube root of the following number by prime factorization method: $10648$

Answer 1

Hint: Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $1$ and which are not the product of any two smaller natural numbers. For Example: $2,{\text{ 3, 5, 7,}}......$ $2$ is the prime number as it can have only $1$ factor. Factors are the number $1$and the number itself. Cube root of any number is the factor which is multiplied with it- self three times.

Complete step by step answer:
To find the prime factorisation, start dividing the given number $10648$by the first prime number,$2$ Continue division until it is not divisible by$2$. Gradually start dividing by $3,{\text{ 5, 7, 11, 13,}}....$
Let’s start dividing the given number $10648$by $2$
$
  10648 \div 2 = 5324 \\
  5324 \div 2 = 2662 \\
  2662 \div 2 = 1331 \\
  1331 \div 2 = 665.5 \\
 $
So, $1331$ is further not divided by$2$. Hence, $10648 = 2 \times 2 \times 2 \times 1331$
Now take, the second prime number, $3$ and divide
$1331 \div 3 = 443.6667$
So, $1331$ is not divisible by$3$. Therefore $3$ is not the factor.
Now, divide $1331{\text{ by 5}}$
$1331 \div 5 = 266.2$
  As $1331$ is not divisible by $5$ and therefore $5$ is not the factor.
Now, divide $1331$ by $7$
$1331 \div 7 = 190.1429$
As, $1331$ is not divisible by$7$. Therefore $7$ is not the factor.
Now, divide $1331$ by $11$
$
  1331 \div 11 = 121 \\
  121 \div 11 = 11 \\
  11 \div 11 = 1 \\
 $
Therefore $11$ is the factor. And $1331 = 11 \times 11 \times 11$
Now, taking the given number-
$10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11$
Taking cube roots both the sides,
\[\sqrt[3]{{10648}} = \sqrt[3]{{2 \times 2 \times 2 \times 11 \times 11 \times 11}}\]
As per the definition we can write –
$
  {2^3} = 2 \times 2 \times 2 \\
  {11^3} = 11 \times 11 \times 11 \\
 $
\[\sqrt[3]{{10648}} = \sqrt[3]{{{2^3} \times {{11}^3}}}\]
As, Cubes and cube-roots cancel each other-
\[
  \sqrt[3]{{10648}} = 2 \times 11 \\
  \sqrt[3]{{10648}} = 22 \\
 \]
Thus, the cube-root of $10648$ is $22$. This is the required solution.

Note: Another method to find the prime factorization is to use the factor tree. A factor tree is the tool that breaks down any number into its prime factors.