GMAT Math : Simplifying Algebraic Expressions

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Example Question #51 : Simplifying Algebraic Expressions

Assume that $x,y \neq 0$ $\displaystyle x,y \neq 0$.

Which of the following expressions is equal to the following expression?

$\frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$ $\displaystyle \frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$

Possible Answers:

$\frac{2 }{x^{2}} + \frac{2 }{y^{2}}$ $\displaystyle \frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

$4x^{2}y^{2}$ $\displaystyle 4x^{2}y^{2}$

$\frac{2 }{x} + \frac{2 }{y}$ $\displaystyle \frac{2 }{x} + \frac{2 }{y}$

$\displaystyle 4$

$\frac{2 }{x^{2}y} + \frac{2 }{xy^{2}}$ $\displaystyle \frac{2 }{x^{2}y} + \frac{2 }{xy^{2}}$

Correct answer:

$\frac{2 }{x^{2}} + \frac{2 }{y^{2}}$ $\displaystyle \frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

Explanation:

$\frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$ $\displaystyle \frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$

$=\frac{(x^{2}+2xy+y^{2}) + (x^{2}-2xy+y^{2}) }{x^{2}y^{2}}$ $\displaystyle =\frac{(x^{2}+2xy+y^{2}) + (x^{2}-2xy+y^{2}) }{x^{2}y^{2}}$

$=\frac{x^{2}+x^{2}+2xy-2xy+y^{2}+y^{2} }{x^{2}y^{2}}$ $\displaystyle =\frac{x^{2}+x^{2}+2xy-2xy+y^{2}+y^{2} }{x^{2}y^{2}}$

$=\frac{2x^{2}+2y^{2} }{x^{2}y^{2}}$ $\displaystyle =\frac{2x^{2}+2y^{2} }{x^{2}y^{2}}$

$=\frac{2x^{2} }{x^{2}y^{2}} + \frac{2y^{2} }{x^{2}y^{2}}$ $\displaystyle =\frac{2x^{2} }{x^{2}y^{2}} + \frac{2y^{2} }{x^{2}y^{2}}$

$=\frac{2 }{y^{2}} + \frac{2 }{x^{2}} = \frac{2 }{x^{2}} + \frac{2 }{y^{2}}$ $\displaystyle =\frac{2 }{y^{2}} + \frac{2 }{x^{2}} = \frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

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GMAT Math : Simplifying Algebraic Expressions

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