Calculus Examples

$36 e^{- 6 \cos^{2} (\frac{π}{6})} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 1

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

$36 e^{- 6 {(\frac{\sqrt{3}}{2})}^{2}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 2

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

$36 e^{- 6 \frac{{\sqrt{3}}^{2}}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 3

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

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Step 3.1

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

$36 e^{- 6 \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 3.2

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

$36 e^{- 6 \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 3.3

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

$36 e^{- 6 \frac{3^{\frac{2}{2}}}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$36 e^{- 6 \frac{3^{\frac{2}{2}}}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 3.4.2

Rewrite the expression.

$36 e^{- 6 \frac{3^{1}}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 3.5

Evaluate the exponent.

$36 e^{- 6 \frac{3}{2^{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 4

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

$36 e^{- 6 (\frac{3}{4})} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $- 6$ .

$36 e^{2 (- 3) \frac{3}{4}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 5.2

Factor $2$ out of $4$ .

$36 e^{2 \cdot - 3 \frac{3}{2 \cdot 2}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 5.3

Cancel the common factor.

$36 e^{2 \cdot - 3 \frac{3}{2 \cdot 2}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 5.4

Rewrite the expression.

$36 e^{- 3 (\frac{3}{2})} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 6

Combine $- 3$ and $\frac{3}{2}$ .

$36 e^{\frac{- 3 \cdot 3}{2}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 7

Simplify the expression.

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Step 7.1

Multiply $- 3$ by $3$ .

$36 e^{\frac{- 9}{2}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 7.2

Move the negative in front of the fraction.

$36 e^{- \frac{9}{2}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$36 \frac{1}{e^{\frac{9}{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 9

Combine $36$ and $\frac{1}{e^{\frac{9}{2}}}$ .

$\frac{36}{e^{\frac{9}{2}}} \sin (\frac{π}{6}) \cos (\frac{π}{6})$

Step 10

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{36}{e^{\frac{9}{2}}} \cdot \frac{1}{2} \cos (\frac{π}{6})$

Step 11

Combine.

$\frac{36 \cdot 1}{e^{\frac{9}{2}} \cdot 2} \cos (\frac{π}{6})$

Step 12

Multiply $36$ by $1$ .

$\frac{36}{e^{\frac{9}{2}} \cdot 2} \cos (\frac{π}{6})$

Step 13

Factor $2$ out of $36$ .

$\frac{2 \cdot 18}{e^{\frac{9}{2}} \cdot 2} \cos (\frac{π}{6})$

Step 14

Cancel the common factors.

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Step 14.1

Factor $2$ out of $e^{\frac{9}{2}} \cdot 2$ .

$\frac{2 \cdot 18}{2 \cdot e^{\frac{9}{2}}} \cos (\frac{π}{6})$

Step 14.2

Cancel the common factor.

$\frac{2 \cdot 18}{2 \cdot e^{\frac{9}{2}}} \cos (\frac{π}{6})$

Step 14.3

Rewrite the expression.

$\frac{18}{e^{\frac{9}{2}}} \cos (\frac{π}{6})$

Step 15

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

$\frac{18}{e^{\frac{9}{2}}} \cdot \frac{\sqrt{3}}{2}$

Step 16

Combine.

$\frac{18 \sqrt{3}}{e^{\frac{9}{2}} \cdot 2}$

Step 17

Factor $2$ out of $18 \sqrt{3}$ .

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

Step 18

Cancel the common factors.

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Step 18.1

Factor $2$ out of $e^{\frac{9}{2}} \cdot 2$ .

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

Step 18.2

Cancel the common factor.

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

Step 18.3

Rewrite the expression.

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

Step 19

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.17317211 \dots$