Question

Apply the properties of integer exponents to generate equivalent expressions to
(4³•7^-²)^-⁵/7⁹•4^-¹³ with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

To generate equivalent expressions with positive exponents, we can use the following properties of integer exponents:

1) Product of Powers Property: (a^m)*(a^n) = a^(m+n)
2) Quotient of Powers Property: (a^m)/(a^n) = a^(m-n)
3) Power of Power Property: (a^m)^n = a^(m*n)
4) Negative Exponent Property: a^(-n) = 1/(a^n)

Let's apply these properties to simplify the expression step by step:

(4³•7^-²)^-⁵/7⁹•4^-¹³

Using the Power of Power Property:
= (4^3)^-5 • (7^-2)^-5 / (7^9) • (4^-13)

Using the Negative Exponent Property:
= 4^(-3*5) • 7^(2*5) / 7^9 • 4^(-13)

Simplifying the exponents:
= 4^(-15) • 7^10 / 7^9 • 4^(-13)

Using the Quotient of Powers Property:
= 4^(-15 - (-13)) • 7^(10-9)

Simplifying further:
= 4^(-15 + 13) • 7^1

= 4^(-2) • 7

Using the Negative Exponent Property:
= 1/(4^2) • 7

= 1/16 • 7

Now, let's multiply and simplify the fraction:

= 7/16

So, the simplified fraction without any remaining exponents is 7/16.