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)
40 ⋅ 5−2(−2)−3=

Answer 1

To simplify the expression \(4^0 \cdot 5^{-2} / (-2)^{-3}\) using properties of integer exponents and to ensure that all exponents are positive, we will follow these steps:

1. Apply \(a^0 = 1\) for any \(a\) that is not zero: \[ 4^0 = 1 \] Therefore, the expression simplifies to: \[ 1 \cdot 5^{-2} / (-2)^{-3} = 5^{-2} / (-2)^{-3} \]

2. Apply the rule \(a^{-n} = 1/a^n\) to rewrite the negative exponents: \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \] and \[ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} \] This allows us to rewrite the entire expression: \[ \frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{25}}{\frac{1}{-8}} = \frac{1}{25} \cdot \left(-8\right) = \frac{-8}{25} \]

Thus, the final simplified expression is: \[ \frac{-8}{25} \]

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