Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

To convert the expression to positive exponents using the properties of integer exponents, we can apply the following rules:

1) For any non-zero number a, a^(-n) is equal to 1/a^n.
2) For any non-zero number a, a^0 is equal to 1.
3) For any non-zero number a, a^m ⋅ a^n is equal to a^(m+n).
4) For any non-zero number a, (a^m)^n is equal to a^(m*n).

Applying these rules, we have:

(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13)

= (43 ⋅ 1/7^2)^(-579) ⋅ 1/4^13

= (43/49)^(-579) ⋅ 1/32768

Now, let's solve the expression:

= (1/(43/49)^579) ⋅ 1/32768

= 1/(1/(43/49)^579 ⋅ 32768)

= (43/49)^579 ⋅ 32768/1

The final result is (43/49)^579 ⋅ 32768.