Mann Ayala
01/19/2024 · Elementary School
Apply the Properties of Integer Exponents to generate equivalent expressions to \( \frac{4^{0} \cdot 5^{-2}}{(-2)^{-3}} \) with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents. (1 point) \( \frac{4^{0} \cdot 5^{-2}}{(-2)^{-3}}=\square \) Check answer Remaining Attempts : 3
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To simplify the expression \( \frac{4^{0} \cdot 5^{-2}}{(-2)^{-3}} \) using the properties of integer exponents, we can follow these steps:
1. **Evaluate \( 4^{0} \)**:
\[
4^{0} = 1
\]
2. **Rewrite \( 5^{-2} \)**:
\[
5^{-2} = \frac{1}{5^{2}} = \frac{1}{25}
\]
3. **Rewrite \( (-2)^{-3} \)**:
\[
(-2)^{-3} = \frac{1}{(-2)^{3}} = \frac{1}{-8} = -\frac{1}{8}
\]
Now, substituting these values back into the original expression:
\[
\frac{4^{0} \cdot 5^{-2}}{(-2)^{-3}} = \frac{1 \cdot \frac{1}{25}}{-\frac{1}{8}} = \frac{\frac{1}{25}}{-\frac{1}{8}}
\]
4. **Dividing by a fraction**:
Dividing by a fraction is the same as multiplying by its reciprocal:
\[
\frac{1}{25} \div -\frac{1}{8} = \frac{1}{25} \cdot -8 = -\frac{8}{25}
\]
Thus, the simplified expression is:
\[
\frac{4^{0} \cdot 5^{-2}}{(-2)^{-3}} = -\frac{8}{25}
\]
So the final answer is:
\[
\boxed{-\frac{8}{25}}
\]
Quick Answer
\[
-\frac{8}{25}
\]
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