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: $$\frac{5^{-2}}{(-2{)}^{-3}}=\frac{\frac{1}{25}}{\frac{1}{-8}}=\frac{1}{25}\cdot (-8)=\frac{-8}{25}$$

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

Therefore, the answer is: $$\boxed{{\displaystyle \frac{-8}{25}}}$$