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# Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product.

Given:

3600

To do:

We have to find the smallest number by which 3600 must be multiplied so that the product is a perfect cube and find the cube root of the product.

Solution:

Prime factorisation of 3600 is,

$3600=2\times2\times2\times2\times3\times3\times5\times5$

$=2^3\times2\times3^2\times5^2$

Grouping the factors in triplets of equal factors, we see that $2, 3^2$ and $5^2$ are left.

In order to make 3600 a perfect cube, we have to multiply it by $2^2\times3\times5=60$.

$3600\times60=2^3\times2\times3^2\times5^2\times2^2\times3\times5$

$=2^3\times2^3\times3^3\times5^3$

$\sqrt[3]{216000}=\sqrt[3]{2^3\times2^3\times3^3\times5^3}$

$=2\times2\times3\times5$

$=60$

The smallest number by which 3600 must be multiplied so that the product is a perfect cube is 60 and the cube root of the product is 60.

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