Find the Exact Value 5sec(pi/4)+7cot(pi/4)

$5 \sec (\frac{π}{4}) + 7 \cot (\frac{π}{4})$

Step 1

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

$5 \frac{2}{\sqrt{2}} + 7 \cot (\frac{π}{4})$

Step 2

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

$5 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) + 7 \cot (\frac{π}{4})$

Step 3

Combine and simplify the denominator.

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

$5 \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} + 7 \cot (\frac{π}{4})$

Raise $\sqrt{2}$ to the power of $1$ .

$5 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + 7 \cot (\frac{π}{4})$

Raise $\sqrt{2}$ to the power of $1$ .

$5 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + 7 \cot (\frac{π}{4})$

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

$5 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + 7 \cot (\frac{π}{4})$

Add $1$ and $1$ .

$5 \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} + 7 \cot (\frac{π}{4})$

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

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

$5 \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + 7 \cot (\frac{π}{4})$

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

$5 \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + 7 \cot (\frac{π}{4})$

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

$5 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + 7 \cot (\frac{π}{4})$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + 7 \cot (\frac{π}{4})$

Rewrite the expression.

$5 \frac{2 \sqrt{2}}{2^{1}} + 7 \cot (\frac{π}{4})$

Evaluate the exponent.

$5 \frac{2 \sqrt{2}}{2} + 7 \cot (\frac{π}{4})$

Step 4

Cancel the common factor of $2$ .

Cancel the common factor.

$5 \frac{2 \sqrt{2}}{2} + 7 \cot (\frac{π}{4})$

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

$5 \sqrt{2} + 7 \cot (\frac{π}{4})$

Step 5

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$5 \sqrt{2} + 7 \cdot 1$

Step 6

Multiply $7$ by $1$ .

$5 \sqrt{2} + 7$

Step 7

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{2} + 7$

Decimal Form:

$14.07106781 \dots$

Find the Exact Value 5sec(pi/4)+7cot(pi/4)

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