Algebra Examples

$(\sqrt{x} + \sqrt{y}) \div \sqrt{x y}$

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Simplify each term.

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Step 7.1

Pull terms out from under the radical.

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

Step 7.2

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

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

Step 7.3

Pull terms out from under the radical.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

Reorder $y$ and $x^{\frac{1}{2}}$ .

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 9

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 10

Move $y^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 11

Simplify the denominator.

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Step 11.1

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.1.1

Move $x^{- \frac{1}{2}}$ .

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

Step 11.1.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 11.1.2.1

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

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

Step 11.1.2.2

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

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

Step 11.1.3

Write $1$ as a fraction with a common denominator.

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

Step 11.1.4

Combine the numerators over the common denominator.

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

Step 11.1.5

Add $- 1$ and $2$ .

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

Step 11.2

Multiply $y$ by $y^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.2.1

Move $y^{- \frac{1}{2}}$ .

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

Step 11.2.2

Multiply $y^{- \frac{1}{2}}$ by $y$ .

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Step 11.2.2.1

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

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

Step 11.2.2.2

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

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

Step 11.2.3

Write $1$ as a fraction with a common denominator.

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

Step 11.2.4

Combine the numerators over the common denominator.

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

Step 11.2.5

Add $- 1$ and $2$ .

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

Step 12

Split the fraction $\frac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x^{\frac{1}{2}} y^{\frac{1}{2}}}$ into two fractions.

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

Step 13

Cancel the common factor.

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

Step 14

Rewrite the expression.

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

Step 15

Cancel the common factor.

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

Step 16

Rewrite the expression.

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