Precalculus Examples

$\frac{\tan (15) + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

Tap for more steps...

Step 1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\tan (45 - 30) + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.2

Separate negation.

$\frac{\tan (45 - (30)) + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.3

Apply the difference of angles identity.

$\frac{\frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.4

The exact value of $\tan (45)$ is $1$ .

$\frac{\frac{1 - \tan (30)}{1 + \tan (45) \tan (30)} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.6

The exact value of $\tan (45)$ is $1$ .

$\frac{\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8

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

Tap for more steps...

Step 1.1.8.1

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

Tap for more steps...

Step 1.1.8.1.1

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

$\frac{\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.1.2

Combine.

$\frac{\frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.2

Apply the distributive property.

$\frac{\frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.1.8.3.1

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

$\frac{\frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.3.2

Cancel the common factor.

$\frac{\frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.3.3

Rewrite the expression.

$\frac{\frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.4

Multiply $3$ by $1$ .

$\frac{\frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.5

Simplify the denominator.

Tap for more steps...

Step 1.1.8.5.1

Multiply $3$ by $1$ .

$\frac{\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.1.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\frac{\frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.5.2.2

Cancel the common factor.

$\frac{\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.5.2.3

Rewrite the expression.

$\frac{\frac{3 - \sqrt{3}}{3 + \sqrt{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.6

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

$\frac{\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.7

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

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.8

Expand the denominator using the FOIL method.

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.9

Simplify.

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.10

Simplify the numerator.

Tap for more steps...

Step 1.1.8.10.1

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

$\frac{\frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.10.2

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

$\frac{\frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.10.3

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

$\frac{\frac{{(3 - \sqrt{3})}^{1 + 1}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.10.4

Add $1$ and $1$ .

$\frac{\frac{{(3 - \sqrt{3})}^{2}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.11

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

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.12

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

Tap for more steps...

Step 1.1.8.12.1

Apply the distributive property.

$\frac{\frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.12.2

Apply the distributive property.

$\frac{\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.12.3

Apply the distributive property.

$\frac{\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13

Simplify and combine like terms.

Tap for more steps...

Step 1.1.8.13.1

Simplify each term.

Tap for more steps...

Step 1.1.8.13.1.1

Multiply $3$ by $3$ .

$\frac{\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 1.1.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.4.3

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.4.4

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.4.5

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.4.6

Add $1$ and $1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 1.1.8.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.5.2

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.5.3

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.8.13.1.5.4.1

Cancel the common factor.

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.5.4.2

Rewrite the expression.

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.1.5.5

Evaluate the exponent.

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.2

Add $9$ and $3$ .

$\frac{\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{\frac{12 - 6 \sqrt{3}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14

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

Tap for more steps...

Step 1.1.8.14.1

Factor $6$ out of $12$ .

$\frac{\frac{6 \cdot 2 - 6 \sqrt{3}}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14.2

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

$\frac{\frac{6 \cdot 2 + 6 (- \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14.3

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

$\frac{\frac{6 (2 - \sqrt{3})}{6} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14.4

Cancel the common factors.

Tap for more steps...

Step 1.1.8.14.4.1

Factor $6$ out of $6$ .

$\frac{\frac{6 (2 - \sqrt{3})}{6 (1)} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14.4.2

Cancel the common factor.

$\frac{\frac{6 (2 - \sqrt{3})}{6 \cdot 1} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14.4.3

Rewrite the expression.

$\frac{\frac{2 - \sqrt{3}}{1} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.1.8.14.4.4

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

$\frac{2 - \sqrt{3} + \tan (15)}{1 - \tan (15) \tan (15)}$

Step 1.2

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

Tap for more steps...

Step 1.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{2 - \sqrt{3} + \tan (45 - 30)}{1 - \tan (15) \tan (15)}$

Step 1.2.2

Separate negation.

$\frac{2 - \sqrt{3} + \tan (45 - (30))}{1 - \tan (15) \tan (15)}$

Step 1.2.3

Apply the difference of angles identity.

$\frac{2 - \sqrt{3} + \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)}}{1 - \tan (15) \tan (15)}$

Step 1.2.4

The exact value of $\tan (45)$ is $1$ .

$\frac{2 - \sqrt{3} + \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)}}{1 - \tan (15) \tan (15)}$

Step 1.2.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 - \sqrt{3} + \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)}}{1 - \tan (15) \tan (15)}$

Step 1.2.6

The exact value of $\tan (45)$ is $1$ .

$\frac{2 - \sqrt{3} + \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)}}{1 - \tan (15) \tan (15)}$

Step 1.2.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 - \sqrt{3} + \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8

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

Tap for more steps...

Step 1.2.8.1

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

Tap for more steps...

Step 1.2.8.1.1

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

$\frac{2 - \sqrt{3} + \frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.1.2

Combine.

$\frac{2 - \sqrt{3} + \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.2

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.8.3.1

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

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.3.2

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.3.3

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.4

Multiply $3$ by $1$ .

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.5

Simplify the denominator.

Tap for more steps...

Step 1.2.8.5.1

Multiply $3$ by $1$ .

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.5.2.2

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.5.2.3

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + \sqrt{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.6

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

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.7

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

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.8

Expand the denominator using the FOIL method.

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.9

Simplify.

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.10

Simplify the numerator.

Tap for more steps...

Step 1.2.8.10.1

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

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.10.2

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

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.10.3

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

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.10.4

Add $1$ and $1$ .

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{2}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.11

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

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.12

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

Tap for more steps...

Step 1.2.8.12.1

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.12.2

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.12.3

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13

Simplify and combine like terms.

Tap for more steps...

Step 1.2.8.13.1

Simplify each term.

Tap for more steps...

Step 1.2.8.13.1.1

Multiply $3$ by $3$ .

$\frac{2 - \sqrt{3} + \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 1.2.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.4.3

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.4.4

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.4.5

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.4.6

Add $1$ and $1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 1.2.8.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.5.2

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.5.3

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.8.13.1.5.4.1

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.5.4.2

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.1.5.5

Evaluate the exponent.

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.2

Add $9$ and $3$ .

$\frac{2 - \sqrt{3} + \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{2 - \sqrt{3} + \frac{12 - 6 \sqrt{3}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14

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

Tap for more steps...

Step 1.2.8.14.1

Factor $6$ out of $12$ .

$\frac{2 - \sqrt{3} + \frac{6 \cdot 2 - 6 \sqrt{3}}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14.2

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

$\frac{2 - \sqrt{3} + \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14.3

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

$\frac{2 - \sqrt{3} + \frac{6 (2 - \sqrt{3})}{6}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14.4

Cancel the common factors.

Tap for more steps...

Step 1.2.8.14.4.1

Factor $6$ out of $6$ .

$\frac{2 - \sqrt{3} + \frac{6 (2 - \sqrt{3})}{6 (1)}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14.4.2

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{6 (2 - \sqrt{3})}{6 \cdot 1}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14.4.3

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{2 - \sqrt{3}}{1}}{1 - \tan (15) \tan (15)}$

Step 1.2.8.14.4.4

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

$\frac{2 - \sqrt{3} + 2 - \sqrt{3}}{1 - \tan (15) \tan (15)}$

Step 1.3

Add $2$ and $2$ .

$\frac{4 - \sqrt{3} - \sqrt{3}}{1 - \tan (15) \tan (15)}$

Step 1.4

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \tan (15) \tan (15)}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

Tap for more steps...

Step 2.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{4 - 2 \sqrt{3}}{1 - \tan (45 - 30) \tan (15)}$

Step 2.1.2

Separate negation.

$\frac{4 - 2 \sqrt{3}}{1 - \tan (45 - (30)) \tan (15)}$

Step 2.1.3

Apply the difference of angles identity.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)} \tan (15)}$

Step 2.1.4

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)} \tan (15)}$

Step 2.1.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)} \tan (15)}$

Step 2.1.6

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)} \tan (15)}$

Step 2.1.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} \tan (15)}$

Step 2.1.8

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

Tap for more steps...

Step 2.1.8.1

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

Tap for more steps...

Step 2.1.8.1.1

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

$\frac{4 - 2 \sqrt{3}}{1 - (\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}) \tan (15)}$

Step 2.1.8.1.2

Combine.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})} \tan (15)}$

Step 2.1.8.2

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15)}$

Step 2.1.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.8.3.1

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15)}$

Step 2.1.8.3.2

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15)}$

Step 2.1.8.3.3

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15)}$

Step 2.1.8.4

Multiply $3$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}} \tan (15)}$

Step 2.1.8.5

Simplify the denominator.

Tap for more steps...

Step 2.1.8.5.1

Multiply $3$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} \tan (15)}$

Step 2.1.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}} \tan (15)}$

Step 2.1.8.5.2.2

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} \tan (15)}$

Step 2.1.8.5.2.3

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \tan (15)}$

Step 2.1.8.6

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

$\frac{4 - 2 \sqrt{3}}{1 - (\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}) \tan (15)}$

Step 2.1.8.7

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})} \tan (15)}$

Step 2.1.8.8

Expand the denominator using the FOIL method.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}} \tan (15)}$

Step 2.1.8.9

Simplify.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.10

Simplify the numerator.

Tap for more steps...

Step 2.1.8.10.1

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.10.2

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6} \tan (15)}$

Step 2.1.8.10.3

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{1 + 1}}{6} \tan (15)}$

Step 2.1.8.10.4

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{2}}{6} \tan (15)}$

Step 2.1.8.11

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.12

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

Tap for more steps...

Step 2.1.8.12.1

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.12.2

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.12.3

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.13

Simplify and combine like terms.

Tap for more steps...

Step 2.1.8.13.1

Simplify each term.

Tap for more steps...

Step 2.1.8.13.1.1

Multiply $3$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 2.1.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6} \tan (15)}$

Step 2.1.8.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} \tan (15)}$

Step 2.1.8.13.1.4.3

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6} \tan (15)}$

Step 2.1.8.13.1.4.4

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6} \tan (15)}$

Step 2.1.8.13.1.4.5

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6} \tan (15)}$

Step 2.1.8.13.1.4.6

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6} \tan (15)}$

Step 2.1.8.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.1.8.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6} \tan (15)}$

Step 2.1.8.13.1.5.2

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6} \tan (15)}$

Step 2.1.8.13.1.5.3

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} \tan (15)}$

Step 2.1.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.8.13.1.5.4.1

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} \tan (15)}$

Step 2.1.8.13.1.5.4.2

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6} \tan (15)}$

Step 2.1.8.13.1.5.5

Evaluate the exponent.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6} \tan (15)}$

Step 2.1.8.13.2

Add $9$ and $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6} \tan (15)}$

Step 2.1.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{12 - 6 \sqrt{3}}{6} \tan (15)}$

Step 2.1.8.14

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

Tap for more steps...

Step 2.1.8.14.1

Factor $6$ out of $12$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 \cdot 2 - 6 \sqrt{3}}{6} \tan (15)}$

Step 2.1.8.14.2

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.14.3

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 (2 - \sqrt{3})}{6} \tan (15)}$

Step 2.1.8.14.4

Cancel the common factors.

Tap for more steps...

Step 2.1.8.14.4.1

Factor $6$ out of $6$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 (2 - \sqrt{3})}{6 (1)} \tan (15)}$

Step 2.1.8.14.4.2

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 (2 - \sqrt{3})}{6 \cdot 1} \tan (15)}$

Step 2.1.8.14.4.3

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{2 - \sqrt{3}}{1} \tan (15)}$

Step 2.1.8.14.4.4

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

$\frac{4 - 2 \sqrt{3}}{1 - (2 - \sqrt{3}) \tan (15)}$

Step 2.2

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 + (- 1 \cdot 2 - - \sqrt{3}) \tan (15)}$

Step 2.3

Multiply $- 1$ by $2$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 - - \sqrt{3}) \tan (15)}$

Step 2.4

Multiply $- - \sqrt{3}$ .

Tap for more steps...

Step 2.4.1

Multiply $- 1$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + 1 \sqrt{3}) \tan (15)}$

Step 2.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \tan (15)}$

Step 2.5

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

Tap for more steps...

Step 2.5.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \tan (45 - 30)}$

Step 2.5.2

Separate negation.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \tan (45 - (30))}$

Step 2.5.3

Apply the difference of angles identity.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)}}$

Step 2.5.4

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)}}$

Step 2.5.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)}}$

Step 2.5.6

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)}}$

Step 2.5.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}}$

Step 2.5.8

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

Tap for more steps...

Step 2.5.8.1

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

Tap for more steps...

Step 2.5.8.1.1

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) (\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}})}$

Step 2.5.8.1.2

Combine.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})}}$

Step 2.5.8.2

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}$

Step 2.5.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.8.3.1

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}$

Step 2.5.8.3.2

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}$

Step 2.5.8.3.3

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}$

Step 2.5.8.4

Multiply $3$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}$

Step 2.5.8.5

Simplify the denominator.

Tap for more steps...

Step 2.5.8.5.1

Multiply $3$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}$

Step 2.5.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}}}$

Step 2.5.8.5.2.2

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}$

Step 2.5.8.5.2.3

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 - \sqrt{3}}{3 + \sqrt{3}}}$

Step 2.5.8.6

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) (\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}})}$

Step 2.5.8.7

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}}$

Step 2.5.8.8

Expand the denominator using the FOIL method.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}}$

Step 2.5.8.9

Simplify.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}$

Step 2.5.8.10

Simplify the numerator.

Tap for more steps...

Step 2.5.8.10.1

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}}$

Step 2.5.8.10.2

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}}$

Step 2.5.8.10.3

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}}$

Step 2.5.8.10.4

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{{(3 - \sqrt{3})}^{2}}{6}}$

Step 2.5.8.11

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}$

Step 2.5.8.12

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

Tap for more steps...

Step 2.5.8.12.1

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}$

Step 2.5.8.12.2

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}$

Step 2.5.8.12.3

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}$

Step 2.5.8.13

Simplify and combine like terms.

Tap for more steps...

Step 2.5.8.13.1

Simplify each term.

Tap for more steps...

Step 2.5.8.13.1.1

Multiply $3$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}$

Step 2.5.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}$

Step 2.5.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}}$

Step 2.5.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 2.5.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}}$

Step 2.5.8.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}$

Step 2.5.8.13.1.4.3

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}}$

Step 2.5.8.13.1.4.4

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}}$

Step 2.5.8.13.1.4.5

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}}$

Step 2.5.8.13.1.4.6

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}}$

Step 2.5.8.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.5.8.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}}$

Step 2.5.8.13.1.5.2

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}}$

Step 2.5.8.13.1.5.3

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}$

Step 2.5.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.8.13.1.5.4.1

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}$

Step 2.5.8.13.1.5.4.2

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}}$

Step 2.5.8.13.1.5.5

Evaluate the exponent.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}}$

Step 2.5.8.13.2

Add $9$ and $3$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}}$

Step 2.5.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{12 - 6 \sqrt{3}}{6}}$

Step 2.5.8.14

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

Tap for more steps...

Step 2.5.8.14.1

Factor $6$ out of $12$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{6 \cdot 2 - 6 \sqrt{3}}{6}}$

Step 2.5.8.14.2

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}}$

Step 2.5.8.14.3

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{6 (2 - \sqrt{3})}{6}}$

Step 2.5.8.14.4

Cancel the common factors.

Tap for more steps...

Step 2.5.8.14.4.1

Factor $6$ out of $6$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{6 (2 - \sqrt{3})}{6 (1)}}$

Step 2.5.8.14.4.2

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{6 (2 - \sqrt{3})}{6 \cdot 1}}$

Step 2.5.8.14.4.3

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{2 - \sqrt{3}}{1}}$

Step 2.5.8.14.4.4

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) (2 - \sqrt{3})}$

Step 2.6

Expand $(- 2 + \sqrt{3}) (2 - \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 2.6.1

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - 2 (2 - \sqrt{3}) + \sqrt{3} (2 - \sqrt{3})}$

Step 2.6.2

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - 2 \cdot 2 - 2 (- \sqrt{3}) + \sqrt{3} (2 - \sqrt{3})}$

Step 2.6.3

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - 2 \cdot 2 - 2 (- \sqrt{3}) + \sqrt{3} \cdot 2 + \sqrt{3} (- \sqrt{3})}$

Step 2.7

Simplify and combine like terms.

Tap for more steps...

Step 2.7.1

Simplify each term.

Tap for more steps...

Step 2.7.1.1

Multiply $- 2$ by $2$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 - 2 (- \sqrt{3}) + \sqrt{3} \cdot 2 + \sqrt{3} (- \sqrt{3})}$

Step 2.7.1.2

Multiply $- 1$ by $- 2$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} (- \sqrt{3})}$

Step 2.7.1.3

Move $2$ to the left of $\sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} (- \sqrt{3})}$

Step 2.7.1.4

Multiply $\sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 2.7.1.4.1

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

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3})}$

Step 2.7.1.4.2

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

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 2.7.1.4.3

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

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{3}}^{1 + 1}}$

Step 2.7.1.4.4

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 2.7.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - {(3^{\frac{1}{2}})}^{2}}$

Step 2.7.1.5.2

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

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}}$

Step 2.7.1.5.3

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

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{2}{2}}}$

Step 2.7.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.7.1.5.4.1

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{2}{2}}}$

Step 2.7.1.5.4.2

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{1}}$

Step 2.7.1.5.5

Evaluate the exponent.

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 1 \cdot 3}$

Step 2.7.1.6

Multiply $- 1$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3}$

Step 2.7.2

Subtract $3$ from $- 4$ .

$\frac{4 - 2 \sqrt{3}}{1 - 7 + 2 \sqrt{3} + 2 \sqrt{3}}$

Step 2.7.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - 7 + 4 \sqrt{3}}$

Step 2.8

Subtract $7$ from $1$ .

$\frac{4 - 2 \sqrt{3}}{- 6 + 4 \sqrt{3}}$

Step 3

Cancel the common factor of $4 - 2 \sqrt{3}$ and $- 6 + 4 \sqrt{3}$ .

Tap for more steps...

Step 3.1

Factor $2$ out of $4$ .

$\frac{2 (2) - 2 \sqrt{3}}{- 6 + 4 \sqrt{3}}$

Step 3.2

Factor $2$ out of $- 2 \sqrt{3}$ .

$\frac{2 (2) + 2 (- \sqrt{3})}{- 6 + 4 \sqrt{3}}$

Step 3.3

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

$\frac{2 (2 - \sqrt{3})}{- 6 + 4 \sqrt{3}}$

Step 3.4

Cancel the common factors.

Tap for more steps...

Step 3.4.1

Factor $2$ out of $- 6$ .

$\frac{2 (2 - \sqrt{3})}{2 \cdot - 3 + 4 \sqrt{3}}$

Step 3.4.2

Factor $2$ out of $4 \sqrt{3}$ .

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

Step 3.4.3

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

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

Step 3.4.4

Cancel the common factor.

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

Step 3.4.5

Rewrite the expression.

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

Step 4

Multiply $\frac{2 - \sqrt{3}}{- 3 + 2 \sqrt{3}}$ by $\frac{- 3 - 2 \sqrt{3}}{- 3 - 2 \sqrt{3}}$ .

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

Step 5

Combine fractions.

Tap for more steps...

Step 5.1

Multiply $\frac{2 - \sqrt{3}}{- 3 + 2 \sqrt{3}}$ by $\frac{- 3 - 2 \sqrt{3}}{- 3 - 2 \sqrt{3}}$ .

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

Step 5.2

Expand the denominator using the FOIL method.

$\frac{(2 - \sqrt{3}) (- 3 - 2 \sqrt{3})}{9 + 6 \sqrt{3} - 6 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 5.3

Simplify.

$\frac{(2 - \sqrt{3}) (- 3 - 2 \sqrt{3})}{- 3}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Expand $(2 - \sqrt{3}) (- 3 - 2 \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

$\frac{2 (- 3 - 2 \sqrt{3}) - \sqrt{3} (- 3 - 2 \sqrt{3})}{- 3}$

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $2$ by $- 3$ .

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

Step 6.2.1.2

Multiply $- 2$ by $2$ .

$\frac{- 6 - 4 \sqrt{3} - \sqrt{3} \cdot - 3 - \sqrt{3} (- 2 \sqrt{3})}{- 3}$

Step 6.2.1.3

Multiply $- 3$ by $- 1$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} - \sqrt{3} (- 2 \sqrt{3})}{- 3}$

Step 6.2.1.4

Multiply $- \sqrt{3} (- 2 \sqrt{3})$ .

Tap for more steps...

Step 6.2.1.4.1

Multiply $- 2$ by $- 1$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \sqrt{3} \sqrt{3}}{- 3}$

Step 6.2.1.4.2

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

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 ({\sqrt{3}}^{1} \sqrt{3})}{- 3}$

Step 6.2.1.4.3

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

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{- 3}$

Step 6.2.1.4.4

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

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 {\sqrt{3}}^{1 + 1}}{- 3}$

Step 6.2.1.4.5

Add $1$ and $1$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 {\sqrt{3}}^{2}}{- 3}$

Step 6.2.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 6.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 {(3^{\frac{1}{2}})}^{2}}{- 3}$

Step 6.2.1.5.2

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

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{\frac{1}{2} \cdot 2}}{- 3}$

Step 6.2.1.5.3

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

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{- 3}$

Step 6.2.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1.5.4.1

Cancel the common factor.

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{- 3}$

Step 6.2.1.5.4.2

Rewrite the expression.

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{1}}{- 3}$

Step 6.2.1.5.5

Evaluate the exponent.

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3}{- 3}$

Step 6.2.1.6

Multiply $2$ by $3$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 6}{- 3}$

Step 6.2.2

Add $- 6$ and $6$ .

$\frac{0 - 4 \sqrt{3} + 3 \sqrt{3}}{- 3}$

Step 6.2.3

Subtract $4 \sqrt{3}$ from $0$ .

$\frac{- 4 \sqrt{3} + 3 \sqrt{3}}{- 3}$

Step 6.2.4

Add $- 4 \sqrt{3}$ and $3 \sqrt{3}$ .

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

Step 7

Dividing two negative values results in a positive value.

$\frac{\sqrt{3}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$