Remove parentheses.
\[{(\frac{{a}^{m}}{{q}^{-n}})}^{m-n}{(\frac{{a}^{n}}{{a}^{-1}})}^{n-1}{(\frac{{q}^{2}}{qm})}^{1-n}=1\]
Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[{(\frac{{a}^{m}}{{q}^{-n}})}^{m-n}{({a}^{n+1})}^{n-1}{(\frac{{q}^{2}}{qm})}^{1-n}=1\]
Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[{(\frac{{a}^{m}}{{q}^{-n}})}^{m-n}{({a}^{n+1})}^{n-1}{({q}^{2-1}{m}^{-1})}^{1-n}=1\]
Simplify \(2-1\) to \(1\).
\[{(\frac{{a}^{m}}{{q}^{-n}})}^{m-n}{({a}^{n+1})}^{n-1}{({q}^{1}{m}^{-1})}^{1-n}=1\]
Use Rule of One: \({x}^{1}=x\).
\[{(\frac{{a}^{m}}{{q}^{-n}})}^{m-n}{({a}^{n+1})}^{n-1}{(q{m}^{-1})}^{1-n}=1\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{{({a}^{m})}^{m-n}}{{({q}^{-n})}^{m-n}}{({a}^{n+1})}^{n-1}{(q{m}^{-1})}^{1-n}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{a}^{m(m-n)}}{{({q}^{-n})}^{m-n}}{({a}^{n+1})}^{n-1}{(q{m}^{-1})}^{1-n}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{a}^{m(m-n)}}{{q}^{-n(m-n)}}{({a}^{n+1})}^{n-1}{(q{m}^{-1})}^{1-n}=1\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\frac{{a}^{m(m-n)}}{\frac{1}{{q}^{n(m-n)}}}{({a}^{n+1})}^{n-1}{(q{m}^{-1})}^{1-n}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{a}^{m(m-n)}}{\frac{1}{{q}^{n(m-n)}}}{a}^{(n+1)(n-1)}{(q{m}^{-1})}^{1-n}=1\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{{a}^{m(m-n)}}{\frac{1}{{q}^{n(m-n)}}}{a}^{(n+1)(n-1)}{q}^{1-n}{({m}^{-1})}^{1-n}=1\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{a}^{m(m-n)}}{\frac{1}{{q}^{n(m-n)}}}{a}^{(n+1)(n-1)}{q}^{1-n}{m}^{-(1-n)}=1\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\frac{{a}^{m(m-n)}}{\frac{1}{{q}^{n(m-n)}}}{a}^{(n+1)(n-1)}{q}^{1-n}\times \frac{1}{{m}^{1-n}}=1\]
Invert and multiply.
\[{a}^{m(m-n)}{q}^{n(m-n)}{a}^{(n+1)(n-1)}{q}^{1-n}\times \frac{1}{{m}^{1-n}}=1\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{{a}^{m(m-n)}{q}^{n(m-n)}{a}^{(n+1)(n-1)}{q}^{1-n}\times 1}{{m}^{1-n}}=1\]
Regroup terms.
\[\frac{{a}^{m(m-n)}{a}^{(n+1)(n-1)}{q}^{n(m-n)}{q}^{1-n}}{{m}^{1-n}}=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{a}^{m(m-n)+(n+1)(n-1)}{q}^{n(m-n)+1-n}}{{m}^{1-n}}=1\]
Simplify \({a}^{m(m-n)+(n+1)(n-1)}{q}^{n(m-n)+1-n}\) to \({a}^{{m}^{2}-mn+{n}^{2}-1}{q}^{n(m-n)+1-n}\).
\[\frac{{a}^{{m}^{2}-mn+{n}^{2}-1}{q}^{n(m-n)+1-n}}{{m}^{1-n}}=1\]
Multiply both sides by \({m}^{1-n}\).
\[{a}^{{m}^{2}-mn+{n}^{2}-1}{q}^{n(m-n)+1-n}={m}^{1-n}\]
Divide both sides by \({q}^{n(m-n)+1-n}\).
\[{a}^{{m}^{2}-mn+{n}^{2}-1}=\frac{{m}^{1-n}}{{q}^{n(m-n)+1-n}}\]
Take the \(({m}^{2}-mn+{n}^{2}-1)\)th root of both sides.
\[a=\sqrt[m^2-mn+n^2-1]{\frac{{m}^{1-n}}{{q}^{n(m-n)+1-n}}}\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[a=\frac{\sqrt[m^2-mn+{n}^{2}-1]{{m}^{1-n}}}{\sqrt[m^2-mn+n^2-1]{{q}^{n(m-n)+1-n}}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[a=\frac{{m}^{\frac{1-n}{{m}^{2}-mn+{n}^{2}-1}}}{\sqrt[m^2-mn+n^2-1]{{q}^{n(m-n)+1-n}}}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[a=\frac{{m}^{\frac{1-n}{{m}^{2}-mn+{n}^{2}-1}}}{{q}^{\frac{n(m-n)+1-n}{{m}^{2}-mn+{n}^{2}-1}}}\]
a=m^((1-n)/(m^2-m*n+n^2-1))/q^((n*(m-n)+1-n)/(m^2-m*n+n^2-1))
