Question

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

Answer 1

Given: $2^x\times 3^y\times 5^z=2160$
First, find out the prime factorisation of 2160.
$\begin{array}{cc}2& 2160\\ 2& 1080\\ 2& 540\\ 2& 270\\ 3& 135\\ 3& 45\\ 3& 15\\ 5& 5\\ & 1\end{array}$
It can be observed that 2160 can be written as $2^4\times 3^3\times 5^1$.
Also,
$$\begin{gathered} 2^x\times 3^y\times 5^z=2^4\times 3^3\times 5^1 \\ \Rightarrow x=4,y=3,z=1 \end{gathered}$$
Therefore, the value of $3^x\times 2^{-y}\times 5^{-z}$ is $3^4\times 2^{-3}\times 5^{-1}=81\times \frac{1}{8}\times \frac{1}{5}=\frac{81}{40}$.