Find the value of \(m\) which satisfies \(\left(\frac{14}{6}\right)^8 \times \left(\frac{6}{14}\right)^3 \times \left(\frac{14}{6}\right)^{10} = \left(\frac{6}{14}\right)^{9m + 9}\).
\(-\frac{8}{3}\)
Write everything in terms of the base \(\frac{6}{14}\). Note that \(\frac{14}{6} = \left(\frac{6}{14}\right)^{-1}\).
So \(\left(\frac{14}{6}\right)^8 = \left(\frac{6}{14}\right)^{-8}\) and \(\left(\frac{14}{6}\right)^{10} = \left(\frac{6}{14}\right)^{-10}\).
Multiplying the left side, add the exponents: \(-8 + 3 + (-10) = -15\), giving \(\left(\frac{6}{14}\right)^{-15}\).
Equating exponents of the same base: \(9m + 9 = -15\).
Then \(9m = -24\), so \(m = -\frac{24}{9} = -\frac{8}{3}\).
Hence, the value of \(m\) is \(-\frac{8}{3}\).
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