Chapter A.3, Problem 58E

Solution

Simplify.

  $-2x$ .

Given:

The given fraction is $\frac{\frac{x^2-y^2}{2xy}}{\frac{y^2-x^2}{4x^{2}y}}$ .

Concept Used:

• Remove any grouping symbol such as brackets and parentheses by multiplying factors.
• Use the exponent rule to remove grouping if the terms are containing exponents.
• Combine the like terms by addition or subtraction.
• Combine the constants.

Calculation:

The given fraction is $\frac{\frac{x^2-y^2}{2xy}}{\frac{y^2-x^2}{4x^{2}y}}$ .

  $\begin{array}{c}\frac{\frac{x^2-y^2}{2xy}}{\frac{y^2-x^2}{4x^{2}y}}=\frac{(x^2-y^2)(4x^{2}y)}{2xy(y^2-x^2)}\\ =\frac{-(y^2-x^2)(4x^{2}y)}{2xy(y^2-x^2)}\\ =\frac{-(4x^{2}y)}{2xy}\\ =-2x\end{array}$

Thus by simplifying $\frac{\frac{x^2-y^2}{2xy}}{\frac{y^2-x^2}{4x^{2}y}}$ to a single fraction it comes $-2x$ .