Let's simplify the given expression step by step.

The expression is:

$\frac{2^4 \times 7^2}{7^6 \times 7 \times 2^{-2}}$

Step 1: Simplify the numerator and denominator

• The numerator is $2^4 \times 7^2$.
• The denominator is $7^6 \times 7^1 \times 2^{-2}$.

Step 2: Combine like terms

• For the powers of $2$:
  • In the numerator: $2^4$
  • In the denominator: $2^{-2}$
• For the powers of $7$:
  • In the numerator: $7^2$
  • In the denominator: $7^6 \times 7^1 = 7^{7}$

The expression now becomes:

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

Step 3: Use the laws of exponents

• Combine the powers of $2$: $2^{4 - (-2)} = 2^{4 + 2} = 2^6$
• Combine the powers of $7$: $7^{2 - 7} = 7^{-5}$

Now the expression is:

$\frac{2^6}{7^5}$

Final Simplified Expression

$\boxed{\frac{64}{16807}}$

This is the fully simplified form of the given expression.

Would you like more details or have any questions?

Related Questions:

1. How would you handle a similar expression with different bases?
2. What are the laws of exponents that you applied in this problem?
3. How do you simplify an expression when the exponents are fractions?
4. What happens if the exponents are negative in both the numerator and denominator?
5. Can you apply the same approach if variables instead of numbers were involved?

Tip:

Always combine like terms and apply exponent rules systematically to simplify expressions efficiently.