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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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