Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Answer 1

To generate equivalent expressions with only positive exponents, we will use the properties of integer exponents:

1) For any nonzero number a, a^(-n) = 1/a^n.
2) For any nonzero number a, a^m * a^n = a^(m + n).
3) For any nonzero number a, a^m / a^n = a^(m - n).

Using these properties, we can simplify the expression:

(4^3 ⋅ 7^(-2))^(-5)/(7^9 ⋅ 4^(-13))

= (4^(3 * -5) ⋅ 7^(-2 * -5))/(7^9 ⋅ 4^(-13))

= (4^(-15) ⋅ 7^10)/(7^9 ⋅ 4^(-13))

= (1/4^15 ⋅ 7^10)/(7^9 ⋅ 1/4^13) (Using property 1)

= 1/(4^15 ⋅ 4^(-13)) ⋅ (7^10/7^9) (Using property 2)

= 1/(4^15/4^(-13)) ⋅ 7^(10-9) (Using property 3)

= 1/(4^15+13) ⋅ 7^1 (Using property 2)

= 1/(4^28) ⋅ 7

Therefore, the simplified expression is 1/(4^28) ⋅ 7, which cannot be further simplified.