Find the Exact Value sec(pi/4)+7csc(pi/6)

$\sec (\frac{π}{4}) + 7 \csc (\frac{π}{6})$

Step 1

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

$\frac{2}{\sqrt{2}} + 7 \csc (\frac{π}{6})$

Step 2

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

$\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + 7 \csc (\frac{π}{6})$

Step 3

Combine and simplify the denominator.

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

$\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} + 7 \csc (\frac{π}{6})$

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + 7 \csc (\frac{π}{6})$

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + 7 \csc (\frac{π}{6})$

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + 7 \csc (\frac{π}{6})$

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} + 7 \csc (\frac{π}{6})$

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

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

$\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + 7 \csc (\frac{π}{6})$

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

$\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + 7 \csc (\frac{π}{6})$

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

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + 7 \csc (\frac{π}{6})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + 7 \csc (\frac{π}{6})$

Rewrite the expression.

$\frac{2 \sqrt{2}}{2^{1}} + 7 \csc (\frac{π}{6})$

Evaluate the exponent.

$\frac{2 \sqrt{2}}{2} + 7 \csc (\frac{π}{6})$

Step 4

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{2}}{2} + 7 \csc (\frac{π}{6})$

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

$\sqrt{2} + 7 \csc (\frac{π}{6})$

Step 5

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

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

Step 6

Multiply $7$ by $2$ .

$\sqrt{2} + 14$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2} + 14$

Decimal Form:

$15.41421356 \dots$

Find the Exact Value sec(pi/4)+7csc(pi/6)

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