Algebra Examples

$\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a b}}$

Step 1

Multiply $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a b}}$ by $\frac{\sqrt{a b}}{\sqrt{a b}}$ .

$\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a b}} \cdot \frac{\sqrt{a b}}{\sqrt{a b}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a b}}$ by $\frac{\sqrt{a b}}{\sqrt{a b}}$ .

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{\sqrt{a b} \sqrt{a b}}$

Step 2.2

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

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{\sqrt{a b}}^{1} \sqrt{a b}}$

Step 2.3

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

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{\sqrt{a b}}^{1} {\sqrt{a b}}^{1}}$

Step 2.4

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

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{\sqrt{a b}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{\sqrt{a b}}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{({(a b)}^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{(a b)}^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{(a b)}^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{(a b)}^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{{(a b)}^{1}}$

Step 2.6.5

Simplify.

$\frac{(\sqrt{a} + \sqrt{b}) \sqrt{a b}}{a b}$

Step 3

Apply the distributive property.

$\frac{\sqrt{a} \sqrt{a b} + \sqrt{b} \sqrt{a b}}{a b}$

Step 4

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

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Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{a (a b)} + \sqrt{b} \sqrt{a b}}{a b}$

Step 4.2

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

$\frac{\sqrt{a^{1} a b} + \sqrt{b} \sqrt{a b}}{a b}$

Step 4.3

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

$\frac{\sqrt{a^{1} a^{1} b} + \sqrt{b} \sqrt{a b}}{a b}$

Step 4.4

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

$\frac{\sqrt{a^{1 + 1} b} + \sqrt{b} \sqrt{a b}}{a b}$

Step 4.5

Add $1$ and $1$ .

$\frac{\sqrt{a^{2} b} + \sqrt{b} \sqrt{a b}}{a b}$

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

$\frac{\sqrt{a^{2} b} + \sqrt{b (a b)}}{a b}$

Step 5.2

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

$\frac{\sqrt{a^{2} b} + \sqrt{a (b^{1} b)}}{a b}$

Step 5.3

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

$\frac{\sqrt{a^{2} b} + \sqrt{a (b^{1} b^{1})}}{a b}$

Step 5.4

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

$\frac{\sqrt{a^{2} b} + \sqrt{a b^{1 + 1}}}{a b}$

Step 5.5

Add $1$ and $1$ .

$\frac{\sqrt{a^{2} b} + \sqrt{a b^{2}}}{a b}$

Step 6

Simplify each term.

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Step 6.1

Pull terms out from under the radical.

$\frac{a \sqrt{b} + \sqrt{a b^{2}}}{a b}$

Step 6.2

Reorder $a$ and $b^{2}$ .

$\frac{a \sqrt{b} + \sqrt{b^{2} a}}{a b}$

Step 6.3

Pull terms out from under the radical.

$\frac{a \sqrt{b} + b \sqrt{a}}{a b}$

Step 7

Simplify the numerator.

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Step 7.1

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

$\frac{a b^{\frac{1}{2}} + b \sqrt{a}}{a b}$

Step 7.2

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

$\frac{a b^{\frac{1}{2}} + b a^{\frac{1}{2}}}{a b}$

Step 7.3

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a b^{\frac{1}{2}} + b a^{\frac{1}{2}}$ .

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Step 7.3.1

Reorder $b$ and $a^{\frac{1}{2}}$ .

$\frac{a b^{\frac{1}{2}} + a^{\frac{1}{2}} b}{a b}$

Step 7.3.2

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a b^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}}) + a^{\frac{1}{2}} b}{a b}$

Step 7.3.3

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b$ .

$\frac{a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} (b^{\frac{1}{2}})}{a b}$

Step 7.3.4

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} (b^{\frac{1}{2}})$ .

$\frac{a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{a b}$

Step 8

Simplify.

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Step 8.1

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

$\frac{b^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{a b a^{- \frac{1}{2}}}$

Step 8.2

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a b a^{- \frac{1}{2}} b^{- \frac{1}{2}}}$

Step 9

Simplify the denominator.

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Step 9.1

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

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Step 9.1.1

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{- \frac{1}{2}} a b \cdot b^{- \frac{1}{2}}}$

Step 9.1.2

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

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Step 9.1.2.1

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{- \frac{1}{2}} a^{1} b \cdot b^{- \frac{1}{2}}}$

Step 9.1.2.2

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{- \frac{1}{2} + 1} b \cdot b^{- \frac{1}{2}}}$

Step 9.1.3

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{- \frac{1}{2} + \frac{2}{2}} b \cdot b^{- \frac{1}{2}}}$

Step 9.1.4

Combine the numerators over the common denominator.

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{- 1 + 2}{2}} b \cdot b^{- \frac{1}{2}}}$

Step 9.1.5

Add $- 1$ and $2$ .

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} b \cdot b^{- \frac{1}{2}}}$

Step 9.2

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

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Step 9.2.1

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} (b^{- \frac{1}{2}} b)}$

Step 9.2.2

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

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Step 9.2.2.1

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} (b^{- \frac{1}{2}} b^{1})}$

Step 9.2.2.2

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} b^{- \frac{1}{2} + 1}}$

Step 9.2.3

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

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} b^{- \frac{1}{2} + \frac{2}{2}}}$

Step 9.2.4

Combine the numerators over the common denominator.

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} b^{\frac{- 1 + 2}{2}}}$

Step 9.2.5

Add $- 1$ and $2$ .

$\frac{a^{\frac{1}{2}} + b^{\frac{1}{2}}}{a^{\frac{1}{2}} b^{\frac{1}{2}}}$