Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 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)
To apply the properties of integer exponents in order to generate equivalent expressions with only positive exponents, we can use the following rules:
1. $a^0=1$ for any non-zero number $a$.
2. $a^{-n}=\frac{1}{a^n}$ for any non-zero number $a$ and positive integer $n$.
Using these rules, we can rewrite the expression $4^0⋅5^{-2}(-2)^{-3}$:
$4^0⋅5^{-2}(-2)^{-3}=(4^0)(\frac{1}{5^2})(\frac{1}{(-2)^3})$
Since any number raised to the power of $0$ is equal to $1$, we have $4^0=1$.
Substituting this value, the expression becomes:
$(1)(\frac{1}{5^2})(\frac{1}{(-2)^3})=(\frac{1}{5^2})(\frac{1}{(-2)^3})$
Since $(-2)^3=(-2)(-2)(-2)=(-2)(4)= -8$, we have:
$(\frac{1}{5^2})(\frac{1}{(-2)^3})=(\frac{1}{5^2})(\frac{1}{-8})$
Simplifying, we have:
$(\frac{1}{5^2})(\frac{1}{-8})=\frac{1}{25} \cdot \frac{1}{-8} = \frac{1}{-200}$
Therefore, the expression $4^0⋅5^{-2}(-2)^{-3}$ simplifies to $\frac{1}{-200}$.