Calculate the Negative Third Power of a Compound Fraction

Let's solve the problem step-by-step:

The given expression is $\left(\frac{4}{9 \times 7}\right)^{-3}$.

Step 1: Apply the negative exponent rule. A negative exponent $-n$ can be transformed by reciprocal and changing the sign of the exponent. Therefore, $\left(\frac{4}{9 \times 7}\right)^{-3} = \left(\frac{9 \times 7}{4}\right)^3$.

Step 2: Express the denominator as a product of integers for clarity: $\frac{9 \times 7}{4}$ is clearer than $\frac{63}{4}$ in context for further steps.

Step 3: Apply the power of a fraction rule. Where $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, therefore, $\left(\frac{9 \times 7}{4}\right)^3 = \frac{(9 \times 7)^3}{4^3}$.

Step 4: Separate the powers within the fraction: The expression $(9 \times 7)^3$ can be expanded using individual exponents: $(9 \times 7)^3 = 9^3 \times 7^3$.

Thus, our expression simplifies to: $\frac{9^3 \times 7^3}{4^3}$.

After analyzing the problem and solving it using the rules of exponents, we conclude that the correct expression is $\frac{9^3 \times 7^3}{4^3}$.

Therefore, the correct choice from the provided options is choice 2: $\frac{9^3 \times 7^3}{4^3}$.