GMAT Practice Question: If n is a positive integer, then (-2^{n})^{-2}+(2^{-n})^{2}=
Question
If n is a positive integer, then (-2^{n})^{-2}+(2^{-n})^{2}=
- 0
- 2^{-2 n}
- 2^{2 n}
- 2^{-2 n+1}
- 2^{2 n+1}
Topics: exponents, factoring, fractions
Solution
Step 1
We simplify the first term by separating the factor -1 and applying the rule for multiplying negative numbers an even number of times.
(-2^n)^{-2} = (-1)^{-2} \times 2^{-2n} = 1 \times 2^{-2n} = 2^{-2n}
Step 2
We simplify the second term by applying the power-of-a-power rule to the exponent.
(2^{-n})^2 = 2^{-2n}
Step 3
We add the two identical simplified terms to find the final expression.
2^{-2n} + 2^{-2n} = 2 \times 2^{-2n} = 2^{-2n + 1}
Answer
2^{-2n + 1}
