Apply the Properties of Integer Exponents to generate equivalent expressions with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Question

Apply the Properties of Integer Exponents to generate equivalent expressions with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
(4^3 * 7^-2)^-5 /7^9 * 4^-13
is the answer 
7/16
or
21/28
or
-7/16
or
16

GPT 3.5 · Answer

To apply the Properties of Integer Exponents, we need to simplify the expression by reducing the exponents and combining like terms.

First, let's simplify the numerator:
(4^3 * 7^-2)^-5

Using the Property of Negative Exponents, we can rewrite 7^-2 as 1/7^2:
(4^3 * 1/7^2)^-5

Simplifying further, we can calculate the value of 4^3 and 1/7^2:
(64 * 1/49)^-5

Now, let's simplify the denominator: 4^-13

Using the Property of Negative Exponents, we can rewrite 4^-13 as 1/4^13.

Now we can substitute these values back into the original expression and simplify:
[(64 * 1/49)^-5] / (7^9 * 1/4^13)

Applying the Property of Exponents (multiplying exponents when dividing with the same base), we get:
[64^-5 * (1/49)^-5] / [7^9 * 4^-13]

Using the Property of Negative Exponents again to eliminate the negative exponents:
[(1/64^5 * 49^5)] / [7^9 * 4^13]

Now, let's simplify each part of the expression:
1/64^5 = 1/1073741824
49^5 = 5764801
7^9 = 40353607
4^13 = 67108864

Substituting these values back into the expression, we get:
[(1/1073741824 * 5764801)] / [40353607 * 67108864]

Simplifying further:
1/1073741824 * 5764801 = 5764801/1073741824

Substituting back into the expression:
[5764801/1073741824] / [40353607 * 67108864]

Now, let's simplify the denominator:
40353607 * 67108864 = 2704549856542448

Substituting back into the expression:
5764801/1073741824 / 2704549856542448

To divide by a fraction, we can multiply by its reciprocal:
5764801/1073741824 * 1/2704549856542448

Simplifying further:
5764801 / (1073741824 * 2704549856542448)

Now, let's simplify the denominator:
1073741824 * 2704549856542448 = 2908088360915081203794299392

Substituting back into the expression:
5764801 / 2908088360915081203794299392

This fraction cannot be simplified any further. It is not one of the answer choices provided, so the expression cannot be solved to a simplified fraction.

Therefore, the answer is none of the options given.