If a real variable x satisfies 3^x^2 = 27 × 9^x, then the value of 2^x^2(2^x)^2 is:

GATE DA 2025 Q9: Exponents: 3^(x²) = 27 × 9^x

If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^{x}$ , then the value of $\dfrac{2^{x^2}}{(2^{x})^{2}}$ is:

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Answer: C (official GATE 2025 final answer key)

Concept used: Equating exponents with a common base.

First step: Write the right side as a power of 3: $27 \times 9^{x} = 3^{3+2x}$ , so $x^2 = 3 + 2x$ . The expression asked for is $2^{x^2 - 2x}$ .

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key