Simplify ( square root of xy)/( square root of x+ square root of y)

$\frac{\sqrt{x y}}{\sqrt{x} + \sqrt{y}}$

Step 1

Multiply $\frac{\sqrt{x y}}{\sqrt{x} + \sqrt{y}}$ by $\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}}$ .

$\frac{\sqrt{x y}}{\sqrt{x} + \sqrt{y}} \cdot \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}}$

Step 2

Multiply $\frac{\sqrt{x y}}{\sqrt{x} + \sqrt{y}}$ by $\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}}$ .

$\frac{\sqrt{x y} (\sqrt{x} - \sqrt{y})}{(\sqrt{x} + \sqrt{y}) (\sqrt{x} - \sqrt{y})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{x y} (\sqrt{x} - \sqrt{y})}{{\sqrt{x}}^{2} - \sqrt{x y} + \sqrt{y x} - {\sqrt{y}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{x y} (\sqrt{x} - \sqrt{y})}{x - y}$

Step 5

Apply the distributive property.

$\frac{\sqrt{x y} \sqrt{x} + \sqrt{x y} (- \sqrt{y})}{x - y}$

Step 6

Multiply $\sqrt{x y} \sqrt{x}$ .

Combine using the product rule for radicals.

$\frac{\sqrt{x y x} + \sqrt{x y} (- \sqrt{y})}{x - y}$

Raise $x$ to the power of $1$ .

$\frac{\sqrt{x^{1} x y} + \sqrt{x y} (- \sqrt{y})}{x - y}$

Raise $x$ to the power of $1$ .

$\frac{\sqrt{x^{1} x^{1} y} + \sqrt{x y} (- \sqrt{y})}{x - y}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{x^{1 + 1} y} + \sqrt{x y} (- \sqrt{y})}{x - y}$

Add $1$ and $1$ .

$\frac{\sqrt{x^{2} y} + \sqrt{x y} (- \sqrt{y})}{x - y}$

Step 7

Simplify terms.

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{x^{2} y} - \sqrt{x y} \sqrt{y}}{x - y}$

Simplify each term.

Pull terms out from under the radical.

$\frac{x \sqrt{y} - \sqrt{x y} \sqrt{y}}{x - y}$

Multiply $- \sqrt{x y} \sqrt{y}$ .

Combine using the product rule for radicals.

$\frac{x \sqrt{y} - \sqrt{y (x y)}}{x - y}$

Raise $y$ to the power of $1$ .

$\frac{x \sqrt{y} - \sqrt{x (y^{1} y)}}{x - y}$

Raise $y$ to the power of $1$ .

$\frac{x \sqrt{y} - \sqrt{x (y^{1} y^{1})}}{x - y}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x \sqrt{y} - \sqrt{x y^{1 + 1}}}{x - y}$

Add $1$ and $1$ .

$\frac{x \sqrt{y} - \sqrt{x y^{2}}}{x - y}$

Reorder $x$ and $y^{2}$ .

$\frac{x \sqrt{y} - \sqrt{y^{2} x}}{x - y}$

Pull terms out from under the radical.

$\frac{x \sqrt{y} - y \sqrt{x}}{x - y}$

Step 8

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{x y^{\frac{1}{2}} - y \sqrt{x}}{x - y}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}{x - y}$

Factor $x^{\frac{1}{2}} y^{\frac{1}{2}}$ out of $x y^{\frac{1}{2}} - y x^{\frac{1}{2}}$ .

Move $y$ .

$\frac{x y^{\frac{1}{2}} - 1 x^{\frac{1}{2}} y}{x - y}$

Factor $x^{\frac{1}{2}} y^{\frac{1}{2}}$ out of $x y^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} y^{\frac{1}{2}} (x^{\frac{1}{2}}) - 1 x^{\frac{1}{2}} y}{x - y}$

Factor $x^{\frac{1}{2}} y^{\frac{1}{2}}$ out of $- 1 x^{\frac{1}{2}} y$ .

$\frac{x^{\frac{1}{2}} y^{\frac{1}{2}} (x^{\frac{1}{2}}) + x^{\frac{1}{2}} y^{\frac{1}{2}} (- y^{\frac{1}{2}})}{x - y}$

Factor $x^{\frac{1}{2}} y^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} y^{\frac{1}{2}} (x^{\frac{1}{2}}) + x^{\frac{1}{2}} y^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

$\frac{x^{\frac{1}{2}} y^{\frac{1}{2}} (x^{\frac{1}{2}} - y^{\frac{1}{2}})}{x - y}$

Simplify ( square root of xy)/( square root of x+ square root of y)

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