Algebra Examples

$\frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}}$

Step 1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.1

Multiply $\frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}}$

Step 1.2

Combine.

$\frac{3 (1 + \frac{\sqrt{3}}{3})}{3 (1 - \frac{\sqrt{3}}{3})}$

Step 2

Apply the distributive property.

$\frac{3 \cdot 1 + 3 \frac{\sqrt{3}}{3}}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Cancel the common factor.

$\frac{3 \cdot 1 + 3 \frac{\sqrt{3}}{3}}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}$

Step 3.1.2

Rewrite the expression.

$\frac{3 \cdot 1 + \sqrt{3}}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{3 \cdot 1 + \sqrt{3}}{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}$

Step 3.2.2

Cancel the common factor.

$\frac{3 \cdot 1 + \sqrt{3}}{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}$

Step 3.2.3

Rewrite the expression.

$\frac{3 \cdot 1 + \sqrt{3}}{3 \cdot 1 - \sqrt{3}}$

Step 4

Simplify.

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Step 4.1

Multiply $3$ by $1$ .

$\frac{3 + \sqrt{3}}{3 \cdot 1 - \sqrt{3}}$

Step 4.2

Multiply $3$ by $1$ .

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

Step 5

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

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

Expand the denominator using the FOIL method.

$\frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 6.3

Simplify.

$\frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $3 + \sqrt{3}$ to the power of $1$ .

$\frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6}$

Step 7.2

Raise $3 + \sqrt{3}$ to the power of $1$ .

$\frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6}$

Step 7.3

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

$\frac{{(3 + \sqrt{3})}^{1 + 1}}{6}$

Step 7.4

Add $1$ and $1$ .

$\frac{{(3 + \sqrt{3})}^{2}}{6}$

Step 8

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}$

Step 9

Expand $(3 + \sqrt{3}) (3 + \sqrt{3})$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

$\frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6}$

Step 9.2

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6}$

Step 9.3

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}$

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}$

Step 10.1.2

Move $3$ to the left of $\sqrt{3}$ .

$\frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6}$

Step 10.1.3

Combine using the product rule for radicals.

$\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6}$

Step 10.1.4

Multiply $3$ by $3$ .

$\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6}$

Step 10.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6}$

Step 10.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6}$

Step 10.2

Add $9$ and $3$ .

$\frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6}$

Step 10.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{12 + 6 \sqrt{3}}{6}$

Step 11

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 11.1

Factor $6$ out of $12$ .

$\frac{6 \cdot 2 + 6 \sqrt{3}}{6}$

Step 11.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{6 \cdot 2 + 6 (\sqrt{3})}{6}$

Step 11.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{6 (2 + \sqrt{3})}{6}$

Step 11.4

Cancel the common factors.

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Step 11.4.1

Factor $6$ out of $6$ .

$\frac{6 (2 + \sqrt{3})}{6 (1)}$

Step 11.4.2

Cancel the common factor.

$\frac{6 (2 + \sqrt{3})}{6 \cdot 1}$

Step 11.4.3

Rewrite the expression.

$\frac{2 + \sqrt{3}}{1}$

Step 11.4.4

Divide $2 + \sqrt{3}$ by $1$ .

$2 + \sqrt{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$