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 use the following rules:

1) (a^m) * (a^n) = a^(m+n)
2) (a^m) / (a^n) = a^(m-n)
3) (a^m)^n = a^(m*n)

Now let's apply these rules to the given expression:

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

Using rule 3, we can simplify the expression inside the parentheses:

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

Now let's substitute this back into the expression:

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

Using rule 2, we can simplify the expression further:

(64/49)^5 / 7^9 * 4^-13 = (64/49)^(5-13) / 7^(9-0) * 4^(-13) = (64/49)^(-8) / 7^9 * 4^(-13)

Using rule 1, we can simplify the expression even more:

(64/49)^(-8) / 7^9 * 4^(-13) = (49/64)^8 / 7^9 * 4^(-13)

Finally, let's rewrite 4^(-13) as (1/4)^13:

(49/64)^8 / 7^9 * (1/4)^13

Now, we can simplify this expression fully by combining the fractions and solving the powers properly:

(49^8 / 64^8) / 7^9 * 1^13 / 4^13 = 5764801 / 37879808 / 7^9 / (2^2)^13
= 5764801 / 37879808 / 7^9 / 2^26
= 5764801 / (37879808 * 7^9) / 2^26

Simplifying the expression further is not possible without specific values for the numbers.