Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13
(
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)
(43 ⋅ 7−2)−579 ⋅ 4−13=
(
4
3

⋅

7
−
2
)
−
5
7
9

⋅

4
−
13
=

Check answerRemaining

Answer 1

To simplify the expression \((4^3 \cdot 7^{-2})^{-579} \cdot 4^{-13}\), we can apply the properties of exponents.

### Step 1: Rewrite the expression
The expression can be clearly written as:
\[
(4^3 \cdot 7^{-2})^{-579} \cdot 4^{-13}
\]

### Step 2: Apply the Power of a Product Property
We need to apply the product inside the parentheses and the property of negative exponents:
\[
(a \cdot b)^{-n} = a^{-n} \cdot b^{-n}
\]

Thus,
\[
(4^3)^{-579} \cdot (7^{-2})^{-579} \cdot 4^{-13}
\]

### Step 3: Simplify each term
We can simplify each term:
\[
(4^3)^{-579} = 4^{-3 \cdot 579} = 4^{-1737}
\]
\[
(7^{-2})^{-579} = 7^{2 \cdot 579} = 7^{1158}
\]

Now the expression becomes:
\[
4^{-1737} \cdot 7^{1158} \cdot 4^{-13}
\]

### Step 4: Combine the powers of 4
Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
\[
4^{-1737 - 13} \cdot 7^{1158} = 4^{-1750} \cdot 7^{1158}
\]

### Step 5: Rewrite with positive exponents
Using the property \(a^{-n} = \frac{1}{a^n}\):
\[
\frac{7^{1158}}{4^{1750}}
\]

### Final Answer
The simplified expression is:
\[
\frac{7^{1158}}{4^{1750}}
\]

So, the final answer is:
\[
\frac{7^{1158}}{4^{1750}}
\] with no remaining exponents.