$(64x^3 \div 27a^{-3})^{-\frac{2}{3}}$

### Concept & Formula This problem tests your mastery of the **Laws of Indices**, specifically dealing with negative exponents and fractional powers. The key exponent rules required are: $$ a^{-m} = \frac{1}{a^m} $$ $$ (\frac{a}{b})^{-m} = (\frac{b}{a})^m $$ $$ (a^m)^n = a^{m \times n} $$ ### Step-by-Step Solution * **Step 1:** Simplify the expression inside the parentheses by removing the negative exponent on the variable $a$. $64x^3 \div 27a^{-3} = \frac{64x^3}{27a^{-3}}$ Since $a^{-3}$ is in the denominator, it moves to the numerator as $a^3$: $\frac{64x^3 a^3}{27}$ * **Step 2:** Apply the negative fractional exponent $-\frac{2}{3}$ to the entire fraction. A negative exponent flips the fraction. $(\frac{64x^3 a^3}{27})^{-\frac{2}{3}} = (\frac{27}{64x^3 a^3})^{\frac{2}{3}}$ * **Step 3:** Evaluate the fractional power $\frac{2}{3}$, which means taking the cube root (denominator $3$) and then squaring the result (numerator $2$). Let's take the cube root of the numerator and denominator first. $\sqrt[3]{27} = 3$ $\sqrt[3]{64x^3 a^3} = 4xa$ So, the expression becomes: $(\frac{3}{4xa})^2$ * **Step 4:** Square both the numerator and the denominator. $\frac{3^2}{(4xa)^2} = \frac{9}{16x^2 a^2}$ ### Exam Strategy & Shortcut To solve this rapidly, handle the constants and variables separately. For the constants: $(64 \div 27)^{-2/3}$. Flip it to get $(27/64)^{2/3}$. The cube root is $3/4$, and squaring it gives $9/16$. This eliminates options (a) and (b). For the variables: $x^3$ and $a^{-3}$ in the denominator becomes $x^3 a^3$. Raising $(x^3 a^3)$ to the power of $-2/3$ directly gives $x^{-2} a^{-2}$, which means they both belong in the denominator squared. This immediately points to option (c). ### Common Pitfall A frequent error is forgetting to apply the exponent to the coefficients (64 and 27) or incorrectly flipping the fraction when dealing with the negative fractional exponent, leading to inverted answers like $\frac{16}{9}$. ### Final Answer **Therefore, the correct answer is $\frac{9}{16x^2 a^2}$.**