apply the properties of integer exponents to generate equivalent expressions to 4^0x 5^-2/(-2)^-3 with only positive exponents then solve the expression. your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Anything to the power of 0 is equal to 1:
4^0 = 1
2. When a negative exponent is moved to the denominator, the sign of the exponent becomes positive:
a^-b = 1/a^b
Using these properties, let's rewrite the given expression step by step:
Expression: 4^0 * 5^-2 / (-2)^-3
Step 1: Apply property 1
4^0 = 1
Expression: 1 * 5^-2 / (-2)^-3
Step 2: Apply property 2
5^-2 = 1/5^2
Expression: 1 * (1/5^2) / (-2)^-3
Step 3: Simplify (-2)^-3
(-2)^-3 = 1/(-2)^3
Expression: 1 * (1/5^2) / (1/(-2)^3)
Step 4: Simplify the resulting expression
1 * (1/5^2) = 1/25
1/(-2)^3 = 1/(-2 * -2 * -2) = 1/(-8) = -1/8
Expression: (1/25) / (-1/8)
Step 5: Divide fractions
(1/25) / (-1/8) = (1/25) * (-8/1) = -8/25
Therefore, the simplified fraction without remaining exponents is -8/25.