Question

If \( x = \left( \frac{3}{2} \right)^2 \times \left( \frac{2}{3} \right)^{-4 \), then the value of \( x^{-2} \) is:}

• A. \( \left( \frac{2}{3} \right)^8 \)
• B. \( \left( \frac{3}{2} \right)^6 \)
• C. \( \left( \frac{3}{2} \right)^{12} \)
• D. \( \left( \frac{2}{3} \right)^{12} \)

Hint:

When dealing with negative exponents, use the rule: \[ \left( \frac{a}{b} \right)^{-n} = \left( \frac{b}{a} \right)^n \] to simplify expressions.

Answer 1

The Correct Option is D

Given: \[ x = \left( \frac{3}{2} \right)^2 \times \left( \frac{2}{3} \right)^{-4} \] Rewriting the negative exponent: \[ x = \left( \frac{3}{2} \right)^2 \times \left( \frac{3}{2} \right)^4 \] Using \( a^m \times a^n = a^{m+n} \): \[ x = \left( \frac{3}{2} \right)^6 \] Now, finding \( x^{-2} \): \[ x^{-2} = \left( \frac{3}{2} \right)^{-12} \] Using the negative exponent rule: \[ x^{-2} = \left( \frac{2}{3} \right)^{12} \]
Thus, the correct answer is \( \left( \frac{2}{3} \right)^{12} \).