(x+1/y)^a * (1/y-x)^a / (y+1/x)^a * (1/x-y)a

asked by guest
on Mar 22, 2025 at 8:52 am



You asked:

Evaluate the expression: (x+1y)a(1y−x)a(y+1x)a⋅(1x−y)⋅a\frac{{\left( x + \frac{1}{y} \right)}^{a} {\left( \frac{1}{y} - x \right)}^{a}}{{\left( y + \frac{1}{x} \right)}^{a}} \cdot \left(\frac{1}{x} - y\right) \cdot a

MathBot Answer:

Evaluated



(x+1y)a(1y−x)a(y+1x)a⋅(1x−y)⋅a=a(−x+1y)a(−y+1x)(x+1y)a(y+1x)−a\displaystyle \frac{{\left( x + \frac{1}{y} \right)}^{a} {\left( \frac{1}{y} - x \right)}^{a}}{{\left( y + \frac{1}{x} \right)}^{a}} \cdot \left(\frac{1}{x} - y\right) \cdot a = a \left(- x + \frac{1}{y}\right)^{a} \left(- y + \frac{1}{x}\right) \left(x + \frac{1}{y}\right)^{a} \left(y + \frac{1}{x}\right)^{- a}

Expanded

(x+1y)a(1y−x)a(y+1x)a⋅(1x−y)⋅a=−ay(−x+1y)a(x+1y)a(y+1x)−a+a(−x+1y)a(x+1y)a(y+1x)−ax\frac{{\left( x + \frac{1}{y} \right)}^{a} {\left( \frac{1}{y} - x \right)}^{a}}{{\left( y + \frac{1}{x} \right)}^{a}} \cdot \left(\frac{1}{x} - y\right) \cdot a = - a y \left(- x + \frac{1}{y}\right)^{a} \left(x + \frac{1}{y}\right)^{a} \left(y + \frac{1}{x}\right)^{- a} + \frac{a \left(- x + \frac{1}{y}\right)^{a} \left(x + \frac{1}{y}\right)^{a} \left(y + \frac{1}{x}\right)^{- a}}{x}

Factored

(x+1y)a(1y−x)a(y+1x)a⋅(1x−y)⋅a=−a(−x+1y)a(x+1y)a(y+1x)−a(xy−1)x\frac{{\left( x + \frac{1}{y} \right)}^{a} {\left( \frac{1}{y} - x \right)}^{a}}{{\left( y + \frac{1}{x} \right)}^{a}} \cdot \left(\frac{1}{x} - y\right) \cdot a = - \frac{a \left(- x + \frac{1}{y}\right)^{a} \left(x + \frac{1}{y}\right)^{a} \left(y + \frac{1}{x}\right)^{- a} \left(x y - 1\right)}{x}

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