Trigonometry Examples

$\frac{\sqrt[4]{3 y}}{\sqrt[4]{8 x^{5}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8 x^{5}$ as $x^{4} \cdot (8 x)$ .

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Step 1.1.1

Factor out $x^{4}$ .

$\frac{\sqrt[4]{3 y}}{\sqrt[4]{8 (x^{4} x)}}$

Step 1.1.2

Reorder $8$ and $x^{4}$ .

$\frac{\sqrt[4]{3 y}}{\sqrt[4]{x^{4} \cdot 8 x}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt[4]{3 y}}{\sqrt[4]{x^{4} \cdot (8 x)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt[4]{3 y}}{x \sqrt[4]{8 x}}$

Step 2

Multiply $\frac{\sqrt[4]{3 y}}{x \sqrt[4]{8 x}}$ by $\frac{{\sqrt[4]{8 x}}^{3}}{{\sqrt[4]{8 x}}^{3}}$ .

$\frac{\sqrt[4]{3 y}}{x \sqrt[4]{8 x}} \cdot \frac{{\sqrt[4]{8 x}}^{3}}{{\sqrt[4]{8 x}}^{3}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[4]{3 y}}{x \sqrt[4]{8 x}}$ by $\frac{{\sqrt[4]{8 x}}^{3}}{{\sqrt[4]{8 x}}^{3}}$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x \sqrt[4]{8 x} {\sqrt[4]{8 x}}^{3}}$

Step 3.2

Move $\sqrt[4]{8 x}$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x (\sqrt[4]{8 x} {\sqrt[4]{8 x}}^{3})}$

Step 3.3

Raise $\sqrt[4]{8 x}$ to the power of $1$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x ({\sqrt[4]{8 x}}^{1} {\sqrt[4]{8 x}}^{3})}$

Step 3.4

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

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {\sqrt[4]{8 x}}^{1 + 3}}$

Step 3.5

Add $1$ and $3$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {\sqrt[4]{8 x}}^{4}}$

Step 3.6

Rewrite ${\sqrt[4]{8 x}}^{4}$ as $8 x$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8 x}$ as ${(8 x)}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {({(8 x)}^{\frac{1}{4}})}^{4}}$

Step 3.6.2

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

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {(8 x)}^{\frac{1}{4} \cdot 4}}$

Step 3.6.3

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

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {(8 x)}^{\frac{4}{4}}}$

Step 3.6.4

Cancel the common factor of $4$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {(8 x)}^{\frac{4}{4}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x {(8 x)}^{1}}$

Step 3.6.5

Simplify.

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x (8 x)}$

Step 4

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1

Move $x$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x \cdot x \cdot 8}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{\sqrt[4]{3 y} {\sqrt[4]{8 x}}^{3}}{x^{2} \cdot 8}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[4]{8 x}}^{3}$ as $\sqrt[4]{{(8 x)}^{3}}$ .

$\frac{\sqrt[4]{3 y} \sqrt[4]{{(8 x)}^{3}}}{x^{2} \cdot 8}$

Step 5.2

Apply the product rule to $8 x$ .

$\frac{\sqrt[4]{3 y} \sqrt[4]{8^{3} x^{3}}}{x^{2} \cdot 8}$

Step 5.3

Raise $8$ to the power of $3$ .

$\frac{\sqrt[4]{3 y} \sqrt[4]{512 x^{3}}}{x^{2} \cdot 8}$

Step 5.4

Rewrite $512 x^{3}$ as $4^{4} \cdot (2 x^{3})$ .

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Step 5.4.1

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{3 y} \sqrt[4]{256 (2) x^{3}}}{x^{2} \cdot 8}$

Step 5.4.2

Rewrite $256$ as $4^{4}$ .

$\frac{\sqrt[4]{3 y} \sqrt[4]{4^{4} \cdot 2 x^{3}}}{x^{2} \cdot 8}$

Step 5.4.3

Add parentheses.

$\frac{\sqrt[4]{3 y} \sqrt[4]{4^{4} \cdot (2 x^{3})}}{x^{2} \cdot 8}$

Step 5.5

Pull terms out from under the radical.

$\frac{\sqrt[4]{3 y} \cdot 4 \sqrt[4]{2 x^{3}}}{x^{2} \cdot 8}$

Step 5.6

Combine exponents.

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Step 5.6.1

Combine using the product rule for radicals.

$\frac{4 \sqrt[4]{3 y (2 x^{3})}}{x^{2} \cdot 8}$

Step 5.6.2

Multiply $2$ by $3$ .

$\frac{4 \sqrt[4]{6 y x^{3}}}{x^{2} \cdot 8}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $4$ and $8$ .

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Step 6.1.1

Factor $4$ out of $4 \sqrt[4]{6 y x^{3}}$ .

$\frac{4 (\sqrt[4]{6 y x^{3}})}{x^{2} \cdot 8}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $4$ out of $x^{2} \cdot 8$ .

$\frac{4 (\sqrt[4]{6 y x^{3}})}{4 (x^{2} \cdot 2)}$

Step 6.1.2.2

Cancel the common factor.

$\frac{4 \sqrt[4]{6 y x^{3}}}{4 (x^{2} \cdot 2)}$

Step 6.1.2.3

Rewrite the expression.

$\frac{\sqrt[4]{6 y x^{3}}}{x^{2} \cdot 2}$

Step 6.2

Move $2$ to the left of $x^{2}$ .

$\frac{\sqrt[4]{6 y x^{3}}}{2 x^{2}}$