Trigonometry Examples

$\sqrt[4]{\frac{2}{125}}$

Step 1

Rewrite $\sqrt[4]{\frac{2}{125}}$ as $\frac{\sqrt[4]{2}}{\sqrt[4]{125}}$ .

$\frac{\sqrt[4]{2}}{\sqrt[4]{125}}$

Step 2

Multiply $\frac{\sqrt[4]{2}}{\sqrt[4]{125}}$ by $\frac{{\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{3}}$ .

$\frac{\sqrt[4]{2}}{\sqrt[4]{125}} \cdot \frac{{\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{3}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[4]{2}}{\sqrt[4]{125}}$ by $\frac{{\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{3}}$ .

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{\sqrt[4]{125} {\sqrt[4]{125}}^{3}}$

Step 3.2

Raise $\sqrt[4]{125}$ to the power of $1$ .

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{1} {\sqrt[4]{125}}^{3}}$

Step 3.3

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

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{1 + 3}}$

Step 3.4

Add $1$ and $3$ .

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{4}}$

Step 3.5

Rewrite ${\sqrt[4]{125}}^{4}$ as $125$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{125}$ as $125^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{{(125^{\frac{1}{4}})}^{4}}$

Step 3.5.2

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

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{125^{\frac{1}{4} \cdot 4}}$

Step 3.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{125^{\frac{4}{4}}}$

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{125^{\frac{4}{4}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{125^{1}}$

Step 3.5.5

Evaluate the exponent.

$\frac{\sqrt[4]{2} {\sqrt[4]{125}}^{3}}{125}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{125}}^{3}$ as $\sqrt[4]{125^{3}}$ .

$\frac{\sqrt[4]{2} \sqrt[4]{125^{3}}}{125}$

Step 4.2

Raise $125$ to the power of $3$ .

$\frac{\sqrt[4]{2} \sqrt[4]{1953125}}{125}$

Step 4.3

Rewrite $1953125$ as $25^{4} \cdot 5$ .

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Step 4.3.1

Factor $390625$ out of $1953125$ .

$\frac{\sqrt[4]{2} \sqrt[4]{390625 (5)}}{125}$

Step 4.3.2

Rewrite $390625$ as $25^{4}$ .

$\frac{\sqrt[4]{2} \sqrt[4]{25^{4} \cdot 5}}{125}$

Step 4.4

Pull terms out from under the radical.

$\frac{\sqrt[4]{2} \cdot 25 \sqrt[4]{5}}{125}$

Step 4.5

Combine exponents.

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Step 4.5.1

Combine using the product rule for radicals.

$\frac{25 \sqrt[4]{2 \cdot 5}}{125}$

Step 4.5.2

Multiply $2$ by $5$ .

$\frac{25 \sqrt[4]{10}}{125}$

Step 5

Cancel the common factor of $25$ and $125$ .

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Step 5.1

Factor $25$ out of $25 \sqrt[4]{10}$ .

$\frac{25 (\sqrt[4]{10})}{125}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $25$ out of $125$ .

$\frac{25 \sqrt[4]{10}}{25 \cdot 5}$

Step 5.2.2

Cancel the common factor.

$\frac{25 \sqrt[4]{10}}{25 \cdot 5}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[4]{10}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{10}}{5}$

Decimal Form:

$0.35565588 \dots$