Trigonometry Examples

$\sqrt[4]{\frac{1}{9}}$

Step 1

Rewrite $1$ as $1^{2}$ .

$\sqrt[4]{\frac{1^{2}}{9}}$

Step 2

Rewrite $9$ as $3^{2}$ .

$\sqrt[4]{\frac{1^{2}}{3^{2}}}$

Step 3

Rewrite $\frac{1^{2}}{3^{2}}$ as ${(\frac{1}{3})}^{2}$ .

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

Step 4

Rewrite $\sqrt[4]{{(\frac{1}{3})}^{2}}$ as $\sqrt{\sqrt{{(\frac{1}{3})}^{2}}}$ .

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

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{\frac{1}{3}}$

Step 6

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\frac{\sqrt{1}}{\sqrt{3}}$

Step 7

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{3}}$

Step 8

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

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

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

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.2

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 9.3

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

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

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Step 9.6.1

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

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$