Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Simplify.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

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

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

Step 8.3

Combine the numerators over the common denominator.

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

Step 8.4

Subtract $1$ from $2$ .

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