GMAT Practice Question: If xy = 1, what is the value of \frac{2^{(x+y)^{2}}}{2^{(x-y)^{2}}}?
Question
If xy = 1, what is the value of \frac{2^{(x+y)^{2}}}{2^{(x-y)^{2}}}?
- 2
- 4
- 8
- 16
- 32
Topics: development formulas, calculations
Solution
Step 1
We apply exponent subtraction to combine the powers of two.
\frac{2^{(x+y)^2}}{2^{(x-y)^2}} = 2^{(x+y)^2 - (x-y)^2}
Step 2
We apply \textbf{\text{"Development Formulas - Squares of +/-"}} to expand each squared binomial.
(x+y)^2 = x^2 + y^2 + 2xy (x-y)^2 = x^2 + y^2 - 2xy
Step 3
We subtract the expanded exponents to simplify the exponent expression.
(x+y)^2 - (x-y)^2 = 4xy
Step 4
We substitute the given value into the exponent and evaluate the power of two.
2^{4xy} = 2^{4 \times 1} = 2^4 = 16
Answer
16
