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 and generate equivalent expressions with only positive exponents, we can use the following rules:
1. Any non-zero number raised to the power of 0 is equal to 1.
2. If a number is raised to a negative exponent, it can be rewritten as the reciprocal of that number raised to the positive exponent.
3. For the product of two numbers with the same base raised to different exponents, we can add the exponents.
4. For the division of two numbers with the same base raised to different exponents, we can subtract the exponent in the denominator from the exponent in the numerator.
Using these rules, we can rewrite the expression:
4^0 ⋅ 5^−2/(−2)^−3
1 ⋅ 5^−2/(−2)^−3
5^−2/((−2)^−3)
Now, let's apply the rules to simplify the expression further:
5^−2 can be rewritten as 1/5^2, since a negative exponent can be written as the reciprocal of the positive exponent:
1/(5^2)/((−2)^−3)
Now, we can simplify further by rewriting (−2)^−3 as 1/(−2)^3:
1/(5^2)/(1/(−2)^3)
Now, let's combine the fractions by multiplying the numerator and denominator by the reciprocal of the second fraction:
1/(5^2) * (−2)^3
Simplifying further, (−2)^3 is equal to −8:
1/(25) * (−8)
Finally, multiplying the fractions gives us:
−8/25
Therefore, the simplified fraction with no remaining exponents is −8/25.