Evaluate the expression. (Simplify your answer fully. Unsimplified or decimal answers will be graded as incorrect.)
[(-1/2)^-2]^3
-Distribute the ^-2 to both the numerator and denominator first
[(-1)^-2/(2)^-2]^3
-Simply the exponents within the parenthesis second
[1/(1/4)]^3
-Simplify within the parenthesis
[4]^3
-simplify
64
To evaluate the expression [(-1/2)^-2]^3, we'll start by simplifying the base and the exponent separately, and then calculate the final value.
1. Simplify the base:
The base of the expression is (-1/2)^-2. To simplify this, we'll first write it as a positive exponent to make it easier to work with. Whenever a negative exponent is present, we can rewrite the expression by taking the reciprocal of the base and changing the exponent to a positive value. So,
(-1/2)^-2 = (1/(-1/2))^2
Now, we can easily simplify the expression within the parentheses by taking the reciprocal:
(1/(-1/2))^2 = (1/[(-1)/2])^2 = (1/(-2/1))^2 = (1/(-2))^2
Next, we square the numerator and denominator separately:
(1/(-2))^2 = (1^2)/((-2)^2) = 1/4
Therefore, the simplified base is 1/4.
2. Simplify the exponent:
The exponent of the expression is 3. Since the base has already been simplified, we can proceed directly to calculating the exponentiation:
(1/4)^3 = 1/4 * 1/4 * 1/4 = 1/64
So, the simplified exponent is 1/64.
3. Calculate the final value:
To evaluate the expression fully, we multiply the simplified base by the simplified exponent:
[(-1/2)^-2]^3 = (1/4)^(1/64)
Hence, the evaluation of the expression [(-1/2)^-2]^3 is equal to 1/4^(1/64). Note that this is already a fully simplified answer.
[(-1/2)^-2]^3
(1/(-1/2)^2)^3 = 1/(1/4)^3 = 1/(1/64)=64