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 ????

Answer 1

To solve the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we'll first apply the properties of integer exponents to rewrite everything with positive exponents before simplifying.

1. **Apply the negative exponent on the numerator:**
\[
(4^3 \cdot 7^{-2})^{-5} = 4^{-15} \cdot 7^{10}
\]
Here, we use the property \((a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}\).
- \(4^3\) raised to \(-5\) becomes \(4^{-15}\) (since \(3 \cdot -5 = -15\)).
- \(7^{-2}\) raised to \(-5\) becomes \(7^{10}\) (since \(-2 \cdot -5 = 10\)).

2. **Rewrite the overall expression:**
Now substituting back, we get:
\[
\frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}}
\]

3. **Combine the terms in the fraction:**
Recall that when dividing exponents with the same base, we subtract the exponents:
\[
= 4^{-15 - (-13)} \cdot 7^{10 - 9} = 4^{-15 + 13} \cdot 7^{1} = 4^{-2} \cdot 7^{1}
\]

4. **Simplify the expression:**
Knowing that \(4^{-2} = \frac{1}{4^2}\), we have:
\[
= \frac{7}{4^2} = \frac{7}{16}
\]

Therefore, the final simplified answer is:
\[
\boxed{\frac{7}{16}}
\]