Find the value of m which satisfies \(\left(\dfrac{26}{7}\right)^5 \times \left(\dfrac{7}{26}\right)^{11} \times \left(\dfrac{26}{7}\right)^{19} = \left(\dfrac{7}{26}\right)^{9m+17}\)
\(-\tfrac{10}{3}\)
Rewrite everything in terms of \(\left(\tfrac{7}{26}\right)\): \(\left(\tfrac{26}{7}\right)^5 = \left(\tfrac{7}{26}\right)^{-5}\) and \(\left(\tfrac{26}{7}\right)^{19} = \left(\tfrac{7}{26}\right)^{-19}\).
Left side exponent (base \(\tfrac{7}{26}\)): \(-5+11-19 = -13\).
So \(\left(\tfrac{7}{26}\right)^{-13} = \left(\tfrac{7}{26}\right)^{9m+17}\), giving \(9m+17=-13\).
\(9m = -30 \Rightarrow m = -\dfrac{30}{9} = -\dfrac{10}{3}\).
Hence, the value of m is \(-\tfrac{10}{3}\).
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