Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Answer 1

To simplify the expression \( \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \) using the properties of integer exponents, we will convert all negative exponents to positive and simplify step by step.

1. Evaluate \( 4^0 \): \[ 4^0 = 1 \]

2. Rewrite the expression: The expression becomes: \[ \frac{1 \cdot 5^{-2}}{(-2)^{-3}} = \frac{5^{-2}}{(-2)^{-3}} \]

3. Convert negative exponents to positive: Recall \( a^{-n} = \frac{1}{a^n} \). So we can rewrite: \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \] and \[ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8} \]

4. Rewrite the original expression again: This gives us: \[ \frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{25}}{-\frac{1}{8}} \]

5. Dividing fractions: To divide by a fraction, you multiply by its reciprocal: \[ \frac{1}{25} \div -\frac{1}{8} = \frac{1}{25} \cdot -8 = -\frac{8}{25} \]

6. Final result: The expression simplifies to: \[ -\frac{8}{25} \]

Therefore, the final answer is: \[ -\frac{8}{25} \]