Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^2)^5\7^9 ⋅ 4^13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

To apply the properties of integer exponents, we can multiply the exponents when raising a power to another power. Using this property, we can rewrite the expression as:

(4^3 ⋅ 7^2)^5 / (7^9 ⋅ 4^13)

We can simplify the numerator and denominator separately.

In the numerator, the expression (4^3 ⋅ 7^2)^5 can be expanded using the property of distributing exponents:

4^(3 * 5) ⋅ 7^(2 * 5)

Simplifying, we get:

4^15 ⋅ 7^10

Similarly, in the denominator, we apply the property:

7^(9 - 9) ⋅ 4^(13 - 13)

Simplifying further:

7^0 ⋅ 4^0

Any nonzero number raised to the power of 0 is equal to 1. Therefore, we have:

1/1

The fraction simplifies to:

1

So, the final answer is 1.