If $2^x \times 3^y \times 5^z = 2160$, find $x, y$ and $z$. Hence, compute the value of $3^x \times 2^{-y} \times 5^{-z}$.

Given:

$2^x \times 3^y \times 5^z = 2160$

To do: 

We have to find $x, y$ and $z$ and compute the value of $3^x \times 2^{-y} \times 5^{-z}$.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

Prime factorisation of 2160 is,

$2160=2^4\times3^3\times5^1$

This implies,

$2^x \times 3^y \times 5^z=2^4\times3^3\times5^1$

Comparing both sides, we get,

$x=4, y=3, z=1$

This implies,

$3^x \times 2^{-y} \times 5^{-z}=3^{4}\times2^{-3}\times5^{-1}$

$=\frac{3^4}{2^3\times5^1}$

$=\frac{81}{8\times5}$

$=\frac{81}{40}$

The values of $x, y$ and $z$ are 4, 3 and 1 respectively. The value of $3^x \times 2^{-y} \times 5^{-z}$ is $\frac{81}{40}$.   

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Simply Easy Learning

Updated on: 10-Oct-2022

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