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Mathematics

Statement 1: (34)−4×(34)−5=(34)3x\Big(\dfrac{3}{4}\Big)^{-4} \times \Big(\dfrac{3}{4}\Big)^{-5} = \Big(\dfrac{3}{4}\Big)^{3x}

⇒ x = -1

Statement 2: (34)−4−5=(34)3x\Big(\dfrac{3}{4}\Big)^{-4 - 5} = \Big(\dfrac{3}{4}\Big)^{3x}

⇒ 3x = -9

  1. Both the statements are true.

  2. Both the statements are false.

  3. Statement 1 is true, and statement 2 is false.

  4. Statement 1 is false, and statement 2 is true.

Indices

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Answer

Given,

⇒(34)−4×(34)−5=(34)3x⇒(34)−4+(−5)=(34)3x⇒(34)−4−5=(34)3x⇒(34)−9=(34)3x⇒3x=−9⇒x=−93⇒x=−3.\Rightarrow \Big(\dfrac{3}{4}\Big)^{-4} \times \Big(\dfrac{3}{4}\Big)^{-5} = \Big(\dfrac{3}{4}\Big)^{3x}\\[1em] \Rightarrow \Big(\dfrac{3}{4}\Big)^{-4 + (-5)} = \Big(\dfrac{3}{4}\Big)^{3x}\\[1em] \Rightarrow \Big(\dfrac{3}{4}\Big)^{-4 - 5} = \Big(\dfrac{3}{4}\Big)^{3x}\\[1em] \Rightarrow \Big(\dfrac{3}{4}\Big)^{-9} = \Big(\dfrac{3}{4}\Big)^{3x}\\[1em] \Rightarrow 3x = -9\\[1em] \Rightarrow x = -\dfrac{9}{3}\\[1em] \Rightarrow x = -3.

∴ Statement 1 is false, and statement 2 is true.

Hence, option 4 is the correct option.

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