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.
To apply the properties of integer exponents, we'll use the following rules:
1. Any non-zero number to the power of 0 is equal to 1.
2. When dividing two numbers with the same base and different exponents, subtract the exponents.
3. The reciprocal of a number to a positive exponent is the same as the number to the negative exponent.
Using these rules, we can simplify the expression:
4^0 ⋅ 5^(-2) ⋅ (-2)^(-3)
Using rule 1, 4^0 is equal to 1. So we're left with:
1 ⋅ 5^(-2) ⋅ (-2)^(-3)
Let's simplify 5^(-2) and (-2)^(-3):
5^(-2) = 1/(5^2) = 1/25
(-2)^(-3) = 1/((-2)^3) = 1/(-8) = -1/8
Now we can substitute these simplified values back into the expression:
1 ⋅ 1/25 ⋅ -1/8
Multiplying these fractions, we get:
(-1)/(25 * 8) = (-1)/200
Therefore, the simplified expression is -1/200.
To apply the Properties of Integer Exponents and generate equivalent expressions with only positive exponents, we can rewrite the given expression step-by-step:
1. Start with 4^0 ⋅ 5^−2(−2)^−3
2. Any number raised to the power of 0 is equal to 1. So, 4^0 = 1.
The expression becomes: 1 ⋅ 5^−2(−2)^−3
3. When we have a negative exponent, we can rewrite it as the reciprocal of the positive exponent.
So, 5^−2 becomes 1/5^2, and (−2)^−3 becomes 1/(−2)^3.
The expression now becomes: 1/(5^2) ⋅ 1/[(−2)^3]
4. Simplifying further, we have 1/25 ⋅ 1/(−8).
5. Multiplying the fractions, we get (1/25) ⨉ (1/−8) = 1/−200.
Therefore, the equivalent expression with only positive exponents is 1/−200.