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.(1 point)

Answer 1

To apply the properties of integer exponents, we can start by simplifying the expression inside the parentheses:

(4^3 ⋅ 7^−2)^−5 = 4^(3*-5) ⋅ 7^(-2*-5) = 4^(-15) ⋅ 7^10

Now, we can rewrite the expression using positive exponents:

(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = (4^(-15) ⋅ 7^10)/(7^9 ⋅ 4^13)

Simplifying further:

= 4^(-15-13) ⋅ 7^(10-9)

= 4^(-28) ⋅ 7^1

Now, to solve the expression, we need to evaluate 4^(-28) and 7^1:

4^(-28) = 1/4^28 (since any number raised to the power of -n is equal to 1 divided by that number raised to the power of n)

7^1 = 7

So the expression becomes:

(1/4^28) ⋅ 7

= 7/4^28