Evaluate ( square root of x- square root of y)/( square root of x+ square root of y)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Simplify the numerator.

Raise $\sqrt{x} - \sqrt{y}$ to the power of $1$ .

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

Raise $\sqrt{x} - \sqrt{y}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Step 4

Rewrite ${(\sqrt{x} - \sqrt{y})}^{2}$ as $(\sqrt{x} - \sqrt{y}) (\sqrt{x} - \sqrt{y})$ .

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

Step 5

Expand $(\sqrt{x} - \sqrt{y}) (\sqrt{x} - \sqrt{y})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

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

Raise $\sqrt{x}$ to the power of $1$ .

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

Raise $\sqrt{x}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

Multiply $- \sqrt{y} (- \sqrt{y})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{y}$ by $1$ .

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

Raise $\sqrt{y}$ to the power of $1$ .

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

Raise $\sqrt{y}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Subtract $\sqrt{x y}$ from $- \sqrt{y x}$ .

Reorder $y$ and $x$ .

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

Subtract $\sqrt{x y}$ from $- \sqrt{x y}$ .

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

Evaluate ( square root of x- square root of y)/( square root of x+ square root of y)

Related Questions

Support LCMGCF.com by disabling your adblocker.