Question

Given the equation: \[ 3^{x^2} = 27 \times 9^x \] Find the value of: \[ \frac{2^{x^2}}{(2^x)^2} \]

• A. 1
• B. 2
• C. 4
• D. 8

Hint:

Exponent rules simplify many algebraic expressions. Always express terms in the same base for easy manipulation.

Answer 1

The Correct Option is A

Step 1: Solve for \( x \) Rewriting the given equation using base 3: \[ 27 = 3^3, \quad 9^x = (3^2)^x = 3^{2x} \] Thus, the equation becomes: \[ 3^{x^2} = 3^3 \times 3^{2x} \] Using exponent addition: \[ 3^{x^2} = 3^{(3 + 2x)} \] Since the bases are the same, we equate exponents: \[ x^2 = 3 + 2x \] Rearrange: \[ x^2 - 2x - 3 = 0 \] Factorizing: \[ (x - 3)(x + 1) = 0 \] Thus, \( x = 3 \) or \( x = -1 \). Step 2: Evaluate the Expression \[ \frac{2^{x^2}}{(2^x)^2} \] Expanding the denominator: \[ (2^x)^2 = 2^{2x} \] So, the given expression simplifies to: \[ \frac{2^{x^2}}{2^{2x}} = 2^{x^2 - 2x} \] Using \( x^2 - 2x = 3 + 2x - 2x = 3 \) (from earlier calculation): \[ 2^0 = 1 \] Conclusion: The correct answer is 1.