Find the Exact Value 2sec(pi/4)+4cot(pi/3)

$2 \sec (\frac{π}{4}) + 4 \cot (\frac{π}{3})$

Step 1

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

$2 \frac{2}{\sqrt{2}} + 4 \cot (\frac{π}{3})$

Step 2

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

$2 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) + 4 \cot (\frac{π}{3})$

Step 3

Combine and simplify the denominator.

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

$2 \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} + 4 \cot (\frac{π}{3})$

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

$2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + 4 \cot (\frac{π}{3})$

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

$2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + 4 \cot (\frac{π}{3})$

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

$2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + 4 \cot (\frac{π}{3})$

Add $1$ and $1$ .

$2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} + 4 \cot (\frac{π}{3})$

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

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

$2 \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + 4 \cot (\frac{π}{3})$

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

$2 \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + 4 \cot (\frac{π}{3})$

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

$2 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + 4 \cot (\frac{π}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + 4 \cot (\frac{π}{3})$

Rewrite the expression.

$2 \frac{2 \sqrt{2}}{2^{1}} + 4 \cot (\frac{π}{3})$

Evaluate the exponent.

$2 \frac{2 \sqrt{2}}{2} + 4 \cot (\frac{π}{3})$

Step 4

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{2 \sqrt{2}}{2} + 4 \cot (\frac{π}{3})$

Rewrite the expression.

$2 \sqrt{2} + 4 \cot (\frac{π}{3})$

Step 5

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$2 \sqrt{2} + 4 \frac{1}{\sqrt{3}}$

Step 6

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

$2 \sqrt{2} + 4 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 7

Combine and simplify the denominator.

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

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{3^{1}}$

Evaluate the exponent.

$2 \sqrt{2} + 4 \frac{\sqrt{3}}{3}$

Step 8

Combine $4$ and $\frac{\sqrt{3}}{3}$ .

$2 \sqrt{2} + \frac{4 \sqrt{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2} + \frac{4 \sqrt{3}}{3}$

Decimal Form:

$5.13782820 \dots$

Find the Exact Value 2sec(pi/4)+4cot(pi/3)

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