Question

Find the smallest number that should be multiplied by the following numbers to get perfect cube. Also find the cube root of the product so obtained.a) 36

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Answer 1

Answer:

Smallest number to be multiplied with 36 to make it a perfect cube is 6 and the cube root of the product so obtained is 6

Answer 2

We have to find

• the smallest number that should be multiplied by 36 to get a perfect cube
• the cube root of the product obtained

We begin with prime factorizing the given number. Below is the prime factorization of 36:

$\Large \begin{array}{c|c} \underline{\sf {2}}&\underline{\sf {\; \; 36 \; \; \: }} \\ \underline{\sf {2}}&\underline{\sf {\; \; 18 \; \; \: }}\\ \underline{\sf {3}}&\underline{\sf {\; \; 9 \; \; \: }}\\ \underline{\sf {3}}&\underline{\sf {\; \; 3 \; \; \: }} \\ & {\sf \; 1 \; \; }\end{array}$

36 = 2 x 2 x 3 x 3

In 36, both factors 2 and 3 occur only two times. A number is a perfect cube only if it has the same factor in a group of three. Thus, to have 2 and 3 in triples, we multiply 36 by 2 x 3 = 6.

36 x 6 = 2 x 2 x 2 x 3 x 3 x 3

216 = (2 x 3)³

$\bf{\sqrt[3]{216}=\:6}$

• Therefore, the smallest number that should be multiplied by 36 to get a perfect cube is 6.
• The product obtained is 216 and its cube root is 6.