To solve the expression using the laws of exponents, we'll break it down step by step.

Problem:

You are asked to find:

$\frac{\left(\frac{4}{7}\right) \times \left(\frac{2^2}{3^2}\right)}{\left(\frac{4^3}{9}\right) \times \left(\frac{4^3}{7^3}\right)}$

Step 1: Simplify the numerator

The numerator is:

$\left(\frac{4}{7}\right) \times \left(\frac{2^2}{3^2}\right) = \frac{4 \times 4}{7 \times 9} = \frac{4 \times 4}{7 \times 9}$

This simplifies to:

$\frac{4 \times 4}{7 \times 9}$

Step 2: Simplify the denominator

The denominator is:

$\frac{4^3}{9} \times \frac{4^3}{7^3} = \frac{4^3 \times 4^3}{9 \times 7^3}$

Step 3: Combine the fractions

Now, the entire expression becomes:

$\frac{\frac{4 \times 4}{7 \times 9}}{\frac{4^3 \times 4^3}{9 \times 7^3}} = \frac{4 \times 4}{7 \times 9} \times \frac{9 \times 7^3}{4^3 \times 4^3}$

Step 4: Simplify using the laws of exponents

Let's simplify each part:

1. Simplify the powers of 4: $\frac{4}{4^3} = \frac{4^1}{4^3} = 4^{1-3} = 4^{-2}$

2. Simplify the powers of 7: $\frac{7}{7^3} = \frac{7^1}{7^3} = 7^{1-3} = 7^{-2}$

So the expression simplifies to:

$4^{-2} \times 7^{-2} \times \frac{9}{9}$

Since $\frac{9}{9} = 1$, we are left with:

$4^{-2} \times 7^{-2}$

Step 5: Convert back to positive exponents

$4^{-2} \times 7^{-2} = \frac{1}{4^2} \times \frac{1}{7^2} = \frac{1}{16} \times \frac{1}{49} = \frac{1}{784}$

Thus, the final simplified expression is:

$\frac{1}{784}$

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you multiply fractions with exponents?
2. What is the difference between negative and positive exponents?
3. Can you explain the law of exponents for multiplying powers with the same base?
4. How do you simplify expressions with mixed exponents?
5. What are the steps to divide fractions with exponents?
6. How do you apply exponents when dealing with fractional bases?
7. What is the best method to simplify complex fractional expressions?
8. How does changing the exponent sign (negative to positive) affect the value?

Tip: Always apply the laws of exponents before performing any other operations to simplify the expression as much as possible.