Calculus Examples

$\tan^{2} (\frac{π}{6}) \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

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

Step 1.3

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

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Step 1.3.1

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{3^{2}} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.3.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{3^{2}} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.3.3

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

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

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

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

Step 1.3.4.2

Rewrite the expression.

$\frac{3^{1}}{3^{2}} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.3.5

Evaluate the exponent.

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

Step 1.4

Raise $3$ to the power of $2$ .

$\frac{3}{9} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.5

Cancel the common factor of $3$ and $9$ .

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Step 1.5.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{9} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 1}{3 \cdot 3} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.5.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 3} \sec (\frac{π}{6}) + \sec^{3} (\frac{π}{6})$

Step 1.5.2.3

Rewrite the expression.

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

Step 1.6

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

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

Step 1.7

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

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

Step 1.8

Combine and simplify the denominator.

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.8.5

Add $1$ and $1$ .

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

Step 1.8.6

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

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Step 1.8.6.1

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

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

Step 1.8.6.2

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

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

Step 1.8.6.3

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

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

Step 1.8.6.4

Cancel the common factor of $2$ .

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Step 1.8.6.4.1

Cancel the common factor.

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

Step 1.8.6.4.2

Rewrite the expression.

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

Step 1.8.6.5

Evaluate the exponent.

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

Step 1.9

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

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Step 1.9.1

Multiply $\frac{1}{3}$ by $\frac{2 \sqrt{3}}{3}$ .

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

Step 1.9.2

Multiply $3$ by $3$ .

$\frac{2 \sqrt{3}}{9} + \sec^{3} (\frac{π}{6})$

Step 1.10

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

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

Step 1.11

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

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

Step 1.12

Combine and simplify the denominator.

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Step 1.12.1

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

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

Step 1.12.2

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

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

Step 1.12.3

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

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

Step 1.12.4

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

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

Step 1.12.5

Add $1$ and $1$ .

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

Step 1.12.6

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

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Step 1.12.6.1

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

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

Step 1.12.6.2

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

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

Step 1.12.6.3

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

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

Step 1.12.6.4

Cancel the common factor of $2$ .

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Step 1.12.6.4.1

Cancel the common factor.

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

Step 1.12.6.4.2

Rewrite the expression.

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

Step 1.12.6.5

Evaluate the exponent.

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

Step 1.13

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.13.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

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

Step 1.13.2

Apply the product rule to $2 \sqrt{3}$ .

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

Step 1.14

Simplify the numerator.

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Step 1.14.1

Raise $2$ to the power of $3$ .

$\frac{2 \sqrt{3}}{9} + \frac{8 {\sqrt{3}}^{3}}{3^{3}}$

Step 1.14.2

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

$\frac{2 \sqrt{3}}{9} + \frac{8 \sqrt{3^{3}}}{3^{3}}$

Step 1.14.3

Raise $3$ to the power of $3$ .

$\frac{2 \sqrt{3}}{9} + \frac{8 \sqrt{27}}{3^{3}}$

Step 1.14.4

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.14.4.1

Factor $9$ out of $27$ .

$\frac{2 \sqrt{3}}{9} + \frac{8 \sqrt{9 (3)}}{3^{3}}$

Step 1.14.4.2

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

$\frac{2 \sqrt{3}}{9} + \frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}}$

Step 1.14.5

Pull terms out from under the radical.

$\frac{2 \sqrt{3}}{9} + \frac{8 \cdot 3 \sqrt{3}}{3^{3}}$

Step 1.14.6

Multiply $8$ by $3$ .

$\frac{2 \sqrt{3}}{9} + \frac{24 \sqrt{3}}{3^{3}}$

Step 1.15

Raise $3$ to the power of $3$ .

$\frac{2 \sqrt{3}}{9} + \frac{24 \sqrt{3}}{27}$

Step 1.16

Cancel the common factor of $24$ and $27$ .

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Step 1.16.1

Factor $3$ out of $24 \sqrt{3}$ .

$\frac{2 \sqrt{3}}{9} + \frac{3 (8 \sqrt{3})}{27}$

Step 1.16.2

Cancel the common factors.

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Step 1.16.2.1

Factor $3$ out of $27$ .

$\frac{2 \sqrt{3}}{9} + \frac{3 (8 \sqrt{3})}{3 (9)}$

Step 1.16.2.2

Cancel the common factor.

$\frac{2 \sqrt{3}}{9} + \frac{3 (8 \sqrt{3})}{3 \cdot 9}$

Step 1.16.2.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{9} + \frac{8 \sqrt{3}}{9}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{2 \sqrt{3} + 8 \sqrt{3}}{9}$

Step 2.2

Add $2 \sqrt{3}$ and $8 \sqrt{3}$ .

$\frac{10 \sqrt{3}}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{10 \sqrt{3}}{9}$

Decimal Form:

$1.92450089 \dots$