Question
Download Solution PDF\(\text{Find the value of } m \text{ which satisfies } \left( \frac{11}{10} \right)^7 \times \left( \frac{10}{11} \right)^{10} \times \left( \frac{11}{10} \right)^9 = \left( \frac{10}{11} \right)^{3m+17}\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(\left( \frac{11}{10} \right)^7 \times \left( \frac{10}{11} \right)^{10} \times \left( \frac{11}{10} \right)^9 = \left( \frac{10}{11} \right)^{3m+17}\)
Formula used:
\(a^m \times a^n = a^{m+n}\) and \(\left( \frac{a}{b} \right)^n = \left( \frac{b}{a} \right)^{-n}\)
Calculation:
Combine powers of same base:
\(\left( \frac{11}{10} \right)^7 \times \left( \frac{11}{10} \right)^9 = \left( \frac{11}{10} \right)^{16}\)
Now the expression becomes:
\(\left( \frac{11}{10} \right)^{16} \times \left( \frac{10}{11} \right)^{10}\)
Write everything in base \(\left( \frac{10}{11} \right)\):
\(\left( \frac{11}{10} \right)^{16} = \left( \frac{10}{11} \right)^{-16}\)
So the full expression becomes:
\(\left( \frac{10}{11} \right)^{-16} \times \left( \frac{10}{11} \right)^{10} = \left( \frac{10}{11} \right)^{-6}\)
Now equating powers:
\(-6 = 3m + 17\)
⇒ \(3m = -6 - 17 = -23\)
⇒ \(m = \frac{-23}{3}\)
∴ The correct answer is \(-\frac{23}{3}\)
Last updated on Jul 17, 2025
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