Question

Pls answer 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.

Answer 1

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

1) Product Rule: (a^m ⋅ b^n)^p = a^(m ⋅ p) ⋅ b^(n ⋅ p)
2) Quotient Rule: (a^m / b^n)^p = a^(m ⋅ p) / b^(n ⋅ p)
3) Power Rule: (a^m)^n = a^(m ⋅ n)

Using the product rule, we can rewrite the expression as:

(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)

Applying the power rule to (4^3 ⋅ 7^(-2))^(-5), we get:

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

Simplifying further, we have:

4^(-15) ⋅ 7^10 / 7^9 ⋅ 4^(-13)

Now, using the quotient rule, we can rewrite the expression as:

(4^(-15) / 4^(-13)) ⋅ (7^10 / 7^9)

Applying the power rule to 4^(-15) and 4^(-13), we get:

(4^(13-15) ⋅ 7^10 / 7^9)

Simplifying further:

4^(-2) ⋅ 7^10 / 7^9

Using the power rule to rewrite 4^(-2) as 1/4^2:

(1/4^2) ⋅ 7^10 / 7^9

Simplifying further:

1/16 ⋅ 7^10 / 7^9

Now, applying the power rule to 7^10 / 7^9, we get:

1/16 ⋅ 7^(10-9)

Simplifying further:

1/16 ⋅ 7^1

Finally, the expression simplifies to:

1/16 ⋅ 7

Which is equal to:

7/16