Simplify: $\frac{(q+\frac{1}{p})^m(q-\frac{1}{p})^m}{(p+\frac{1}{q})^m(p-\frac{1}{q})^m}$

Given:

$\frac{(q+\frac{1}{p})^m(q-\frac{1}{p})^m}{(p+\frac{1}{q})^m(p-\frac{1}{q})^m}$

To do:

We have to simplify the given expression.
Solution:

$\frac{(q+\frac{1}{p})^m(q-\frac{1}{p})^m}{(p+\frac{1}{q})^m(p-\frac{1}{q})^m}$

$ \begin{array}{l}
=\frac{\left(\frac{pq+1}{p}\right)^{m}\left(\frac{pq-1}{p}\right)^{m}}{\left(\frac{pq+1}{q}\right)^{m}\left(\frac{pq-1}{q}\right)^{m}}\\
\\
=\frac{\frac{( pq+1)^{m}( pq-1)^{m}}{p^{2m}}}{\frac{( pq+1)^{m}( pq-1)^{m}}{q^{2m}}} \times \frac{q^{2m}}{p^{2m}}\\
\\
=\left(\frac{q}{p}\right)^{2m}\\
\end{array}$. 

Therefore,

$\frac{(q+\frac{1}{p})^m(q-\frac{1}{p})^m}{(p+\frac{1}{q})^m(p-\frac{1}{q})^m}=(\frac{q}{p})^{2m}$.

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Updated on: 10-Oct-2022

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