Find the value of \(m\) which satisfies \(\left(\frac{23}{5}\right)^{13} \times \left(\frac{5}{23}\right)^{9} \times \left(\frac{23}{5}\right)^{5} = \left(\frac{5}{23}\right)^{6m + 9}\).
\(-3\)
Combine the two powers of \(\frac{23}{5}\) by adding exponents: \(\left(\frac{23}{5}\right)^{13} \times \left(\frac{23}{5}\right)^{5} = \left(\frac{23}{5}\right)^{18}\).
Now the left side is \(\left(\frac{23}{5}\right)^{18} \times \left(\frac{5}{23}\right)^{9}\). Writing \(\left(\frac{5}{23}\right)^{9} = \left(\frac{23}{5}\right)^{-9}\), it becomes \(\left(\frac{23}{5}\right)^{18 - 9} = \left(\frac{23}{5}\right)^{9}\).
Express this with the same base as the right side: \(\left(\frac{23}{5}\right)^{9} = \left(\frac{5}{23}\right)^{-9}\).
Equating exponents of \(\frac{5}{23}\): \(6m + 9 = -9\).
So \(6m = -18\), giving \(m = -3\).
Hence, the value of \(m\) is \(-3\).
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