Calculus Examples

$\frac{3 (\cos^{5} (\frac{π}{4}))}{5}$

Step 1

Simplify the numerator.

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Step 1.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{3 {(\frac{\sqrt{2}}{2})}^{5}}{5}$

Step 1.2

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

$\frac{3 \frac{{\sqrt{2}}^{5}}{2^{5}}}{5}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$\frac{3 \frac{\sqrt{2^{5}}}{2^{5}}}{5}$

Step 1.3.2

Raise $2$ to the power of $5$ .

$\frac{3 \frac{\sqrt{32}}{2^{5}}}{5}$

Step 1.3.3

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.3.3.1

Factor $16$ out of $32$ .

$\frac{3 \frac{\sqrt{16 (2)}}{2^{5}}}{5}$

Step 1.3.3.2

Rewrite $16$ as $4^{2}$ .

$\frac{3 \frac{\sqrt{4^{2} \cdot 2}}{2^{5}}}{5}$

Step 1.3.4

Pull terms out from under the radical.

$\frac{3 \frac{4 \sqrt{2}}{2^{5}}}{5}$

Step 1.4

Raise $2$ to the power of $5$ .

$\frac{3 \frac{4 \sqrt{2}}{32}}{5}$

Step 1.5

Cancel the common factor of $4$ and $32$ .

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Step 1.5.1

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

$\frac{3 \frac{4 (\sqrt{2})}{32}}{5}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $4$ out of $32$ .

$\frac{3 \frac{4 \sqrt{2}}{4 \cdot 8}}{5}$

Step 1.5.2.2

Cancel the common factor.

$\frac{3 \frac{4 \sqrt{2}}{4 \cdot 8}}{5}$

Step 1.5.2.3

Rewrite the expression.

$\frac{3 \frac{\sqrt{2}}{8}}{5}$

Step 2

Combine $3$ and $\frac{\sqrt{2}}{8}$ .

$\frac{\frac{3 \sqrt{2}}{8}}{5}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 \sqrt{2}}{8} \cdot \frac{1}{5}$

Step 4

Multiply $\frac{3 \sqrt{2}}{8} \cdot \frac{1}{5}$ .

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Step 4.1

Multiply $\frac{3 \sqrt{2}}{8}$ by $\frac{1}{5}$ .

$\frac{3 \sqrt{2}}{8 \cdot 5}$

Step 4.2

Multiply $8$ by $5$ .

$\frac{3 \sqrt{2}}{40}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{2}}{40}$

Decimal Form:

$0.10606601 \dots$