Question

Find the cube root of the following numbers by prime factorization.
a. $110592$ b. $54872$

Answer 1

(a) 110592

Prime factors of 110592-

$110592=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3$

110592 can be written as a product of its prime factors in the following way:

$110592=2^{12}.3^3$

$\therefore \sqrt[3]{3110592}=\sqrt[3]{32^{12}\times 3^3}={(2^{12}\times 3^3)}^{\frac{1}{3}}$

$\Rightarrow =2^{12\times \frac{1}{3}}.3^{3\times \frac{1}{3}}$

$\Rightarrow =2^4\times 3$

$\therefore \sqrt[3]{3110592}=2^4\times 3=2\times 2\times 2\times 2\times 3=16\times 3=48$

Hence, cube root of 110592 is 48.

(b) 54872

Prime factors of 54872-

$54872=2\times 2\times 2\times 19\times 19\times 19$

54872 can be written as a product of its prime factors in the following way:

$54872=2^3.{19}^3$

$\therefore \sqrt[3]{354872}=\sqrt[3]{32^3\times {19}^3}={(2^3\times {19}^3)}^{\frac{1}{3}}$

$\Rightarrow =2^{3\times \frac{1}{3}}.{19}^{3\times \frac{1}{3}}$

$\Rightarrow =2\times 19$

$\therefore \sqrt[3]{354872}=2\times 19=38$

Hence, cube root of 54872 is 38.