Question

Simply (2√x -3√y) / (√x + √y ) to its simplest form

Answer 1

To simplify the expression \(\frac{2\sqrt{x} - 3\sqrt{y}}{\sqrt{x} + \sqrt{y}}\), we can use the method of multiplying the numerator and the denominator by the conjugate of the denominator.

The conjugate of the denominator \(\sqrt{x} + \sqrt{y}\) is \(\sqrt{x} - \sqrt{y}\). Therefore, we multiply the numerator and denominator by this conjugate:

\[
\frac{2\sqrt{x} - 3\sqrt{y}}{\sqrt{x} + \sqrt{y}} \cdot \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}} = \frac{(2\sqrt{x} - 3\sqrt{y})(\sqrt{x} - \sqrt{y})}{(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})}
\]

Now, simplifying the denominator:

\[
(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = \sqrt{x}^2 - \sqrt{y}^2 = x - y
\]

Next, we simplify the numerator:

\[
(2\sqrt{x} - 3\sqrt{y})(\sqrt{x} - \sqrt{y}) = 2\sqrt{x} \cdot \sqrt{x} - 2\sqrt{x} \cdot \sqrt{y} - 3\sqrt{y} \cdot \sqrt{x} + 3\sqrt{y} \cdot \sqrt{y}
\]

This simplifies to:

\[
2x - 2\sqrt{xy} - 3\sqrt{xy} + 3y = 2x - 5\sqrt{xy} + 3y
\]

Putting it all together, we have:

\[
\frac{2x - 5\sqrt{xy} + 3y}{x - y}
\]

Thus, the simplified form of the expression is:

\[
\frac{2x - 5\sqrt{xy} + 3y}{x - y}
\]