Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (4 to the power of 3 ⋅ 7 to the power of −2) to the power of −5 / 7 to the power of 9 ⋅ 4 to the power of −13
with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(43 ⋅ 7−2)−579 ⋅ 4−13=

Answer 1

To simplify the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we'll start by applying the properties of integer exponents.

### Step 1: Simplify the Numerator
We begin with the numerator \((4^3 \cdot 7^{-2})^{-5}\):
\[
(4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5}
\]
Using the property \((a^m)^n = a^{m \cdot n}\):
\[
= 4^{-15} \cdot 7^{10}
\]

### Step 2: Rewrite the Expression
Now substituting back into the expression, we have:
\[
\frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}}
\]

### Step 3: Apply the Quotient Rule
Using the quotient rule of exponents, \(a^m / a^n = a^{m - n}\):
\[
= \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9}
\]
Calculating each term:
\[
= 4^{-15 - (-13)} \cdot 7^{10 - 9}
\]
This simplifies to:
\[
= 4^{-2} \cdot 7^1
\]

### Step 4: Rewrite with Positive Exponents
To express \(4^{-2}\) with positive exponents:
\[
4^{-2} = \frac{1}{4^2} = \frac{1}{16}
\]
Thus, we rewrite our expression:
\[
= \frac{1}{16} \cdot 7 = \frac{7}{16}
\]

### Final Answer
So, the simplified fraction with no remaining exponents is:
\[
\boxed{\frac{7}{16}}
\]