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

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

7/16
or
21/28
or
-7/16
or
16

Answer 1

To apply the properties of integer exponents, we can first simplify the expression within the parentheses:

4^3 = 4 * 4 * 4 = 64
7^-2 = 1 / 7^2 = 1 / 49

Therefore, (4^3 * 7^-2)^-5 = (64 * (1 / 49))^-5 = (64/49)^-5.

Similarly, we can simplify the expression 7^9 * 4^-13:

4^-13 = 1 / 4^13
Therefore, 7^9 * 4^-13 = 7^9 * (1 / 4^13) = 7^9 / 4^13.

Now, we can substitute these simplified expressions back into the original expression:

(4^3 * 7^-2)^-5 / 7^9 * 4^-13 = ((64/49)^-5) / (7^9 / 4^13).

To simplify further, we can invert the denominator and multiply:

((64/49)^-5) / (7^9 / 4^13) = ((64/49)^-5) * (4^13 / 7^9).

Now, let's simplify each fraction:

(64/49)^-5 = (49/64)^5 = 49^5 / 64^5
4^13 = 4^9 * 4^4 = 65536
7^9 = 7^5 * 7^4 = 16807 * 2401

Substituting these values back into the expression:

((64/49)^-5) * (4^13 / 7^9) = (49^5 / 64^5) * (65536 / (16807 * 2401)).

Now, let's simplify each side separately:

49^5 = 49 * 49 * 49 * 49 * 49 = 5764801
64^5 = 64 * 64 * 64 * 64 * 64 = 1073741824
65536 / (16807 * 2401) = 65536 / (40194707) ≈ 0.001630780

Substituting these values back into the expression:

(49^5 / 64^5) * (65536 / (16807 * 2401)) = (5764801 / 1073741824) * 0.001630780.

Multiplying the fractions:

(5764801 / 1073741824) * 0.001630780 ≈ 0.000000309999.

Therefore, the simplified fraction is approximately 0.00000031, which cannot be further simplified.