Simplify ( square root of a+2 square root of y)/( square root of a-2 square root of y)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

Simplify the numerator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{a} \sqrt{a}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

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

Multiply $2$ by $2$ .

$\frac{a + 2 \sqrt{y a} + 2 \sqrt{a y} + 4 \sqrt{y} \sqrt{y}}{a - 4 y}$

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$\frac{a + 2 \sqrt{y a} + 2 \sqrt{a y} + 4 y}{a - 4 y}$

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

Reorder $y$ and $a$ .

$\frac{a + 2 \sqrt{a y} + 2 \sqrt{a y} + 4 y}{a - 4 y}$

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

$\frac{a + 4 \sqrt{a y} + 4 y}{a - 4 y}$

Simplify ( square root of a+2 square root of y)/( square root of a-2 square root of y)

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