Find the smallest number by which 14,580 must be multiplied to make a perfect cube. Also, find the cube root of the perfect cube number obtained.

Finding prime factors of 14580

$14580 = 2\times 2 \times (3 \times 3 \times 3) \times (3 \times 3 \times 3) \times 5$

Since the prime factor 2 and 5 are not in triplets,

Hence, 14,580 must be multiplied with 2 x 5 x 5 = 50

$14580 \times 50 = 729000\\[1em] \sqrt[3]{729000} = \sqrt[3]{(2 \times 2\times 2) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3) \times (5 \times 5 \times 5)}\\[1em] \sqrt[3]{729000} = \sqrt[3]{2 \times 3 \times 3 \times 50}\\[1em] \sqrt[3]{729000} = 90$

14,580 should be multiplied with 50 so that the product is a perfect cube. The cube root of 729000 is 90.