Question

Find the cube root by prime factorization of 10648.

Answer 1

Hint- To find the cube root of a number by factorization, first, find the prime factors of the number and make a group of triplets of the same numbers from the prime factors and then find their products. For example, the prime factor of \[\left( c \right) = a \times a \times b \times b \times a \times b = \underline {\left[ {a \times a \times a} \right]} \times \underline {\left[ {b \times b \times b} \right]} = a \times b\]
The prime number is the number which is either divisible by 1 or by itself. (e.g. 2, 3, 5, 7, 11).


Complete step by step solution:
Let’s find the cube root of the number using the factorization method first we will factorize the given numbers only by the prime numbers.

\[
   2\underline {\left| {10648} \right.} \\
   2\underline {\left| {5324} \right.} \\
   2\underline {\left| {2662} \right.} \\
   11\underline {\left| {1331} \right.} \\
   11\underline {\left| {121} \right.} \\
   11\underline {\left| {11} \right.} \\
   \underline {\left| 1 \right.} \\
 \]
Hence we can write \[\left( {10648} \right) = 2 \times 2 \times 2 \times 11 \times 11 \times 11\]
Now make triplet group of the factors of 10648:

\[
  \left( {10648} \right) = \left[ {2 \times 2 \times 2} \right] \times \left[ {11 \times 11 \times 11} \right] \\
  \sqrt[3]{{\left( {10648} \right)}} = 2 \times 11 = 22 \\
 \]
Hence the cube root of 10648 is 22.

Additional Information: To check whether 22 is the cube root of 10648 or not we will find the cube of the number,
\[{\left( {22} \right)^3} = 22 \times 22 \times 22 = 10648\]
Hence we can say 22 is the cube root of 10648.

Note: The cube root of a number can either be found by using the estimation method or by the factorization method. But the best and easy method of finding cube roots is the factorization method as this has fewer calculations and saves time as well.