Question

Find the cube root of a given number by prime factorization method. 15625.

Answer 1

Hint: To find the cube root of a number using prime factorization method, we will first resolve the number into the product of primes and then from each cubic power of a number, we will select one and then multiply the selected numbers.

Complete step-by-step answer:
The method of prime factorization is used to “break down” or express a given number as a product of prime numbers, if a prime number occurs more than once in the factorization, it is usually expressed in exponential forms. The steps to find the cube root of a number is as shown.
Step I: We will first resolve the given number into prime factors. The prime factors of 15625 are
$\begin{align}
  & 5\left| \!{\underline {\,
  15625 \,}} \right. \\
 & 5\left| \!{\underline {\,
  3125 \,}} \right. \\
 & 5\left| \!{\underline {\,
  625 \,}} \right. \\
 & 5\left| \!{\underline {\,
  125 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1 \\
\end{align}$
Therefore, we can say that, $15625=5\times 5\times 5\times 5\times 5\times 5$ .
Step II: We will group the factors in 3 in such a way that each number of the group is the same. Thus, after grouping, we will get:
\[15625=\left( 5\times 5\times 5 \right)\times \left( 5\times 5\times 5 \right).\]
Step III: we will now take one factor from each group. Thus, we will get the factor 5 and 5.
Step IV: Now, we will find the product of the factors obtained in step III. The product we will get is the required cube root of the number. Thus the cube root of the number 15625 will be.
Cube root of 15625 $=5\times 5$
$=25.$
In other words, we can say that $\sqrt{15625}=25$ .

Note: The prime factorization method is valid only when the given number is a perfect cube root of an integer. In case, the given number is not a perfect cube root then we have to apply some other methods like longhand methods.