Trigonometry Examples

$\sqrt[4]{\frac{13}{25}}$

Step 1

Rewrite $\sqrt[4]{\frac{13}{25}}$ as $\frac{\sqrt[4]{13}}{\sqrt[4]{25}}$ .

$\frac{\sqrt[4]{13}}{\sqrt[4]{25}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt[4]{13}}{\sqrt[4]{5^{2}}}$

Step 2.2

Rewrite $\sqrt[4]{5^{2}}$ as $\sqrt{\sqrt{5^{2}}}$ .

$\frac{\sqrt[4]{13}}{\sqrt{\sqrt{5^{2}}}}$

Step 2.3

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

$\frac{\sqrt[4]{13}}{\sqrt{5}}$

Step 3

Multiply $\frac{\sqrt[4]{13}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt[4]{13}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt[4]{13}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt[4]{13} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt[4]{13} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt[4]{13} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 4.4

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

$\frac{\sqrt[4]{13} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{\sqrt[4]{13} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{13} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{\sqrt[4]{13} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{\sqrt[4]{13} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{13} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{13} \sqrt{5}}{5^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{\sqrt[4]{13} \sqrt{5}}{5}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite the expression using the least common index of $4$ .

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{13} \cdot 5^{\frac{1}{2}}}{5}$

Step 5.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{13} \cdot 5^{\frac{2}{4}}}{5}$

Step 5.1.3

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

$\frac{\sqrt[4]{13} \sqrt[4]{5^{2}}}{5}$

Step 5.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{13 \cdot 5^{2}}}{5}$

Step 5.3

Raise $5$ to the power of $2$ .

$\frac{\sqrt[4]{13 \cdot 25}}{5}$

Step 6

Multiply $13$ by $25$ .

$\frac{\sqrt[4]{325}}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{325}}{5}$

Decimal Form:

$0.84918210 \dots$