Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

Step 1.3

To write $\sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.4

Combine $\sqrt{3}$ and $\frac{3}{3}$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Rewrite $\frac{\sqrt{3} \cdot 3 - \sqrt{3}}{3}$ in a factored form.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

Step 2.3

Multiply $\sqrt{3} \frac{\sqrt{3}}{3}$ .

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Step 2.3.1

Combine $\sqrt{3}$ and $\frac{\sqrt{3}}{3}$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Evaluate the exponent.

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

Step 2.5

Divide $3$ by $3$ .

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

Step 2.6

Add $1$ and $1$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.57735026 \dots$