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Question: If \(x^{y} = y^{x},\) then \((x/y)^{(x/y)} = x^{(x/y) - k},\) where \(k =\)...

If xy=yx,x^{y} = y^{x}, then (x/y)(x/y)=x(x/y)−k,(x/y)^{(x/y)} = x^{(x/y) - k}, where k=k =

A

0

B

1

C

–1

D

None of these

Answer

1

Explanation

Solution

xy=yx⇒(xy)1/x=yx^{y} = y^{x} \Rightarrow (x^{y})^{1/x} = y

Now, (xy)x/y=(xxy/x)x/y=(x1−yx)x/y\left( \frac{x}{y} \right)^{x/y} = \left( \frac{x}{x^{y/x}} \right)^{x/y} = \left( x^{1 - \frac{y}{x}} \right)^{x/y}

=x(x/y)−1=x(x/y)−k⇒k=1x^{(x/y) - 1} = x^{(x/y) - k} \Rightarrow k = 1.