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GATE 2025 CY – Question 9

General Aptitude · Quantitative Aptitude: Arithmetic and Number Computation · 2 marks · Multiple choice

If a real variable $x$ satisfies $3^{x^2}=27\times9^x$, then the value of $\frac{2^{x^2}}{(2^x)^2}$ is:

  1. $2^{-1}$
  2. $2^0$
  3. $2^3$
  4. $2^{15}$

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Correct answer: (C) $2^3$

Explanation

**Step 1: write the right side as a power of 3.** $27=3^3$ and $9^x=3^{2x}$, so
$$3^{x^2}=3^3\cdot3^{2x}=3^{\,3+2x}.$$

**Step 2: equate the exponents.**
$$x^2=3+2x\;\Rightarrow\;x^2-2x=3 .$$

**Step 3: the required expression.**
$$\frac{2^{x^2}}{(2^x)^2}=\frac{2^{x^2}}{2^{2x}}=2^{\,x^2-2x}=2^3 .$$

Answer **$2^3$** (C).