Algebra Examples

$\frac{3 + \sqrt{b}}{3 \sqrt{b}}$

Step 1

Multiply $\frac{3 + \sqrt{b}}{3 \sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{3 + \sqrt{b}}{3 \sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{3 + \sqrt{b}}{3 \sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 \sqrt{b} \sqrt{b}}$

Step 2.2

Move $\sqrt{b}$ .

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 (\sqrt{b} \sqrt{b})}$

Step 2.3

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

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 ({\sqrt{b}}^{1} \sqrt{b})}$

Step 2.4

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

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 ({\sqrt{b}}^{1} {\sqrt{b}}^{1})}$

Step 2.5

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

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 {\sqrt{b}}^{1 + 1}}$

Step 2.6

Add $1$ and $1$ .

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 {\sqrt{b}}^{2}}$

Step 2.7

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

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Step 2.7.1

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

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 {(b^{\frac{1}{2}})}^{2}}$

Step 2.7.2

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

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 b^{\frac{1}{2} \cdot 2}}$

Step 2.7.3

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

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 b^{\frac{2}{2}}}$

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 b^{\frac{2}{2}}}$

Step 2.7.4.2

Rewrite the expression.

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 b^{1}}$

Step 2.7.5

Simplify.

$\frac{(3 + \sqrt{b}) \sqrt{b}}{3 b}$

Step 3

Apply the distributive property.

$\frac{3 \sqrt{b} + \sqrt{b} \sqrt{b}}{3 b}$

Step 4

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

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Step 4.1

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

$\frac{3 \sqrt{b} + {\sqrt{b}}^{1} \sqrt{b}}{3 b}$

Step 4.2

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

$\frac{3 \sqrt{b} + {\sqrt{b}}^{1} {\sqrt{b}}^{1}}{3 b}$

Step 4.3

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

$\frac{3 \sqrt{b} + {\sqrt{b}}^{1 + 1}}{3 b}$

Step 4.4

Add $1$ and $1$ .

$\frac{3 \sqrt{b} + {\sqrt{b}}^{2}}{3 b}$

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

$\frac{3 \sqrt{b} + {(b^{\frac{1}{2}})}^{2}}{3 b}$

Step 5.2

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

$\frac{3 \sqrt{b} + b^{\frac{1}{2} \cdot 2}}{3 b}$

Step 5.3

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

$\frac{3 \sqrt{b} + b^{\frac{2}{2}}}{3 b}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{3 \sqrt{b} + b^{\frac{2}{2}}}{3 b}$

Step 5.4.2

Rewrite the expression.

$\frac{3 \sqrt{b} + b^{1}}{3 b}$

Step 5.5

Simplify.

$\frac{3 \sqrt{b} + b}{3 b}$

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

$\frac{3 b^{\frac{1}{2}} + b}{3 b}$

Step 6.2

Factor $b^{\frac{1}{2}}$ out of $3 b^{\frac{1}{2}} + b$ .

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Step 6.2.1

Factor $b^{\frac{1}{2}}$ out of $3 b^{\frac{1}{2}}$ .

$\frac{b^{\frac{1}{2}} \cdot 3 + b}{3 b}$

Step 6.2.2

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

$\frac{b^{\frac{1}{2}} \cdot 3 + b^{1}}{3 b}$

Step 6.2.3

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

$\frac{b^{\frac{1}{2}} \cdot 3 + b^{\frac{1}{2}} b^{\frac{1}{2}}}{3 b}$

Step 6.2.4

Factor $b^{\frac{1}{2}}$ out of $b^{\frac{1}{2}} \cdot 3 + b^{\frac{1}{2}} b^{\frac{1}{2}}$ .

$\frac{b^{\frac{1}{2}} (3 + b^{\frac{1}{2}})}{3 b}$

Step 7

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

$\frac{3 + b^{\frac{1}{2}}}{3 b \cdot b^{- \frac{1}{2}}}$

Step 8

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

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Step 8.1

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

$\frac{3 + b^{\frac{1}{2}}}{3 (b^{- \frac{1}{2}} b)}$

Step 8.2

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

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Step 8.2.1

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

$\frac{3 + b^{\frac{1}{2}}}{3 (b^{- \frac{1}{2}} b^{1})}$

Step 8.2.2

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

$\frac{3 + b^{\frac{1}{2}}}{3 b^{- \frac{1}{2} + 1}}$

Step 8.3

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

$\frac{3 + b^{\frac{1}{2}}}{3 b^{- \frac{1}{2} + \frac{2}{2}}}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{3 + b^{\frac{1}{2}}}{3 b^{\frac{- 1 + 2}{2}}}$

Step 8.5

Add $- 1$ and $2$ .

$\frac{3 + b^{\frac{1}{2}}}{3 b^{\frac{1}{2}}}$