Find the value of \(m\) which satisfies \(\left(\frac{28}{8}\right)^{17} \times \left(\frac{8}{28}\right)^{20} \times \left(\frac{28}{8}\right)^{3} = \left(\frac{8}{28}\right)^{9m + 14}\).
\(-\frac{14}{9}\)
Combine the two powers of \(\frac{28}{8}\) on the left by adding exponents: \(\left(\frac{28}{8}\right)^{17} \times \left(\frac{28}{8}\right)^{3} = \left(\frac{28}{8}\right)^{20}\).
The left side is now \(\left(\frac{28}{8}\right)^{20} \times \left(\frac{8}{28}\right)^{20}\), and since \(\frac{28}{8}\) and \(\frac{8}{28}\) are reciprocals, this product equals \(1\).
Write 1 as a power of the base on the right: \(1 = \left(\frac{8}{28}\right)^{0}\).
Equating exponents of the same base gives \(9m + 14 = 0\).
Solve for \(m\): \(9m = -14 \Rightarrow m = -\frac{14}{9}\).
Hence, the value of \(m\) is \(-\frac{14}{9}\).
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