Find the Exact Value cot(arccos(-( square root of 3)/3))

$\cot (\arccos (- \frac{\sqrt{3}}{3}))$

Step 1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2

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

$- \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3

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

$- \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4

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

$- \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5

Add $1$ and $1$ .

$- \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6

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

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

$- \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$- \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$- \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$- \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$- \frac{\sqrt{2}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{2}}{2}$

Decimal Form:

$- 0.70710678 \dots$