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Mathematics

If (56)5×(65)4=(56)3x\Big(\dfrac{5}{6}\Big)^5 \times \Big(\dfrac{6}{5}\Big)^{-4} = \Big(\dfrac{5}{6}\Big)^{3x}, then the value of xx is:

  1. 13\dfrac{1}{3}

  2. 203\dfrac{20}{3}

  3. -3

  4. 3

Exponents

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Answer

(56)5×(65)4=(56)3x(56)5×(56)4=(56)3x(56)5+4=(56)3x(56)9=(56)3x9=3xx=93x=3\Big(\dfrac{5}{6}\Big)^5 \times \Big(\dfrac{6}{5}\Big)^{-4} = \Big(\dfrac{5}{6}\Big)^{3x}\\[1em] \Rightarrow \Big(\dfrac{5}{6}\Big)^5 \times \Big(\dfrac{5}{6}\Big)^4 = \Big(\dfrac{5}{6}\Big)^{3x}\\[1em] \Rightarrow \Big(\dfrac{5}{6}\Big)^{5+4} = \Big(\dfrac{5}{6}\Big)^{3x}\\[1em] \Rightarrow \Big(\dfrac{5}{6}\Big)^9 = \Big(\dfrac{5}{6}\Big)^{3x}\\[1em] \Rightarrow 9 = 3x\\[1em] \Rightarrow x = \dfrac{9}{3}\\[1em] \Rightarrow x = 3

Hence, option 4 is the correct option.

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