Question

Perform all indicated operations, and write each answer with positive integer exponents. [ (x-2 + y-2)/ (x-2 - y-2) ] * [ (x+y)/(x-y) ]

Accepted Answer

Start by rewriting the expression clearly: $$\left( \frac{x$$^{-2}$$ + y$$^{-2}$$}{x$$^{-2}$$ - y$$^{-2}$$} ight) 	imes \left( \frac{x + y}{x - y} ight). $$Recognize that $$x^{-2}$$ = \frac{1}{$$x^2$$}$$ and y$$^{-2}$$ = \frac{1}{y^2$}. $$Rewrite the numerator and denominator of the first fraction using positive exponents: $$\frac{\frac{1}{x^2$} + \frac{1}{y^2$}}{\frac{1}{x^2$} - \frac{1}{y^2$}}. $$Find a common denominator for both the numerator and denominator of the large fraction: For the numerator, combine $$\frac{1}{x^2$} + \frac{1}{y^2$} as$$ $$\frac{y^2$ + x^2$}{x^2$ y^2$}$$; for the denominator, combine $$\frac{1}{x^2$} - \frac{1}{y^2$} as$$ $$\frac{y^2$ - x^2$}{x^2$ y^2$}. $$Rewrite the large fraction as $$\frac{\frac{y^2$ + x^2$}{x^2$ y^2$}}{\frac{y^2$ - x^2$}{x^2$ y^2$}}. $$Since both numerator and denominator share the same denominator $$x^2$$ $$y^2$$, simplify by multiplying the numerator by the reciprocal of the denominator: $$\frac{y^2$ + x^2$}{x^2$ y^2$} 	imes \frac{x^2$ y^2$}{y^2$ - x^2$} = \frac{y^2$ + x^2$}{y^2$ - x^2$}. $$Now multiply this result by the second fraction $$\frac{x + y}{x - y}$$: $$\frac{y^2$ + x^2$}{y^2$ - x^2$} 	imes \frac{x + y}{x - y}. $$Notice that $$y^2$$ - $$x^2$$ can be factored as $$(y - x)(y + x). $$Use this factorization to simplify the expression further, and rewrite all terms with positive exponents.

Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Laws of Exponents

Algebraic Fractions and Simplification

Difference of Squares