Algebra Examples

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

Step 1

Multiply $\frac{x + 2 \sqrt{y}}{x - 2 \sqrt{y}}$ by $\frac{x + 2 \sqrt{y}}{x + 2 \sqrt{y}}$ .

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

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Multiply $\frac{x + 2 \sqrt{y}}{x - 2 \sqrt{y}}$ by $\frac{x + 2 \sqrt{y}}{x + 2 \sqrt{y}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} + x (2 \sqrt{y}) - 2 \sqrt{y} x - 4 {\sqrt{y}}^{2}}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

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

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 {(y^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 y^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 y^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.4.1

Cancel the common factor.

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 y^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 y^{1}}$

Step 2.3.5

Simplify.

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 y}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Raise $x + 2 \sqrt{y}$ to the power of $1$ .

$\frac{{(x + 2 \sqrt{y})}^{1} (x + 2 \sqrt{y})}{x^{2} - 4 y}$

Step 3.2

Raise $x + 2 \sqrt{y}$ to the power of $1$ .

$\frac{{(x + 2 \sqrt{y})}^{1} {(x + 2 \sqrt{y})}^{1}}{x^{2} - 4 y}$

Step 3.3

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

$\frac{{(x + 2 \sqrt{y})}^{1 + 1}}{x^{2} - 4 y}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(x + 2 \sqrt{y})}^{2}}{x^{2} - 4 y}$

Step 4

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

$\frac{(x + 2 \sqrt{y}) (x + 2 \sqrt{y})}{x^{2} - 4 y}$

Step 5

Expand $(x + 2 \sqrt{y}) (x + 2 \sqrt{y})$ using the FOIL Method.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\frac{x (x + 2 \sqrt{y}) + 2 \sqrt{y} (x + 2 \sqrt{y})}{x^{2} - 4 y}$

Step 5.2

Apply the distributive property.

$\frac{x \cdot x + x (2 \sqrt{y}) + 2 \sqrt{y} (x + 2 \sqrt{y})}{x^{2} - 4 y}$

Step 5.3

Apply the distributive property.

$\frac{x \cdot x + x (2 \sqrt{y}) + 2 \sqrt{y} x + 2 \sqrt{y} (2 \sqrt{y})}{x^{2} - 4 y}$

Step 6

Simplify and combine like terms.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x (2 \sqrt{y}) + 2 \sqrt{y} x + 2 \sqrt{y} (2 \sqrt{y})}{x^{2} - 4 y}$

Step 6.1.2

Rewrite using the commutative property of multiplication.

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 2 \sqrt{y} (2 \sqrt{y})}{x^{2} - 4 y}$

Step 6.1.3

Multiply $2 \sqrt{y} (2 \sqrt{y})$ .

Tap for more steps...

Step 6.1.3.1

Multiply $2$ by $2$ .

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 \sqrt{y} \sqrt{y}}{x^{2} - 4 y}$

Step 6.1.3.2

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

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 ({\sqrt{y}}^{1} \sqrt{y})}{x^{2} - 4 y}$

Step 6.1.3.3

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

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 ({\sqrt{y}}^{1} {\sqrt{y}}^{1})}{x^{2} - 4 y}$

Step 6.1.3.4

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

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 {\sqrt{y}}^{1 + 1}}{x^{2} - 4 y}$

Step 6.1.3.5

Add $1$ and $1$ .

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 {\sqrt{y}}^{2}}{x^{2} - 4 y}$

Step 6.1.4

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

Tap for more steps...

Step 6.1.4.1

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

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 {(y^{\frac{1}{2}})}^{2}}{x^{2} - 4 y}$

Step 6.1.4.2

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

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 y^{\frac{1}{2} \cdot 2}}{x^{2} - 4 y}$

Step 6.1.4.3

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

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 y^{\frac{2}{2}}}{x^{2} - 4 y}$

Step 6.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.4.4.1

Cancel the common factor.

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 y^{\frac{2}{2}}}{x^{2} - 4 y}$

Step 6.1.4.4.2

Rewrite the expression.

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 y^{1}}{x^{2} - 4 y}$

Step 6.1.4.5

Simplify.

$\frac{x^{2} + 2 x \sqrt{y} + 2 \sqrt{y} x + 4 y}{x^{2} - 4 y}$

Step 6.2

Reorder the factors of $2 \sqrt{y} x$ .

$\frac{x^{2} + 2 x \sqrt{y} + 2 x \sqrt{y} + 4 y}{x^{2} - 4 y}$

Step 6.3

Add $2 x \sqrt{y}$ and $2 x \sqrt{y}$ .

$\frac{x^{2} + 4 x \sqrt{y} + 4 y}{x^{2} - 4 y}$