Trigonometry Examples

$\frac{\sqrt[3]{5 x}}{\sqrt[3]{4 y^{7}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 y^{7}$ as ${(y^{2})}^{3} \cdot (4 y)$ .

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Step 1.1.1

Factor out $y^{6}$ .

$\frac{\sqrt[3]{5 x}}{\sqrt[3]{4 (y^{6} y)}}$

Step 1.1.2

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$\frac{\sqrt[3]{5 x}}{\sqrt[3]{4 ({(y^{2})}^{3} y)}}$

Step 1.1.3

Reorder $4$ and ${(y^{2})}^{3}$ .

$\frac{\sqrt[3]{5 x}}{\sqrt[3]{{(y^{2})}^{3} \cdot 4 y}}$

Step 1.1.4

Add parentheses.

$\frac{\sqrt[3]{5 x}}{\sqrt[3]{{(y^{2})}^{3} \cdot (4 y)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt[3]{5 x}}{y^{2} \sqrt[3]{4 y}}$

Step 2

Multiply $\frac{\sqrt[3]{5 x}}{y^{2} \sqrt[3]{4 y}}$ by $\frac{{\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{2}}$ .

$\frac{\sqrt[3]{5 x}}{y^{2} \sqrt[3]{4 y}} \cdot \frac{{\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{2}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[3]{5 x}}{y^{2} \sqrt[3]{4 y}}$ by $\frac{{\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{2}}$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} \sqrt[3]{4 y} {\sqrt[3]{4 y}}^{2}}$

Step 3.2

Move $\sqrt[3]{4 y}$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} (\sqrt[3]{4 y} {\sqrt[3]{4 y}}^{2})}$

Step 3.3

Raise $\sqrt[3]{4 y}$ to the power of $1$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} ({\sqrt[3]{4 y}}^{1} {\sqrt[3]{4 y}}^{2})}$

Step 3.4

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

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {\sqrt[3]{4 y}}^{1 + 2}}$

Step 3.5

Add $1$ and $2$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {\sqrt[3]{4 y}}^{3}}$

Step 3.6

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

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 y}$ as ${(4 y)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {({(4 y)}^{\frac{1}{3}})}^{3}}$

Step 3.6.2

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

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {(4 y)}^{\frac{1}{3} \cdot 3}}$

Step 3.6.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {(4 y)}^{\frac{3}{3}}}$

Step 3.6.4

Cancel the common factor of $3$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {(4 y)}^{\frac{3}{3}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} {(4 y)}^{1}}$

Step 3.6.5

Simplify.

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{2} (4 y)}$

Step 4

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 4.1

Move $y$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y \cdot y^{2} \cdot 4}$

Step 4.2

Multiply $y$ by $y^{2}$ .

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Step 4.2.1

Raise $y$ to the power of $1$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{1} y^{2} \cdot 4}$

Step 4.2.2

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

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{1 + 2} \cdot 4}$

Step 4.3

Add $1$ and $2$ .

$\frac{\sqrt[3]{5 x} {\sqrt[3]{4 y}}^{2}}{y^{3} \cdot 4}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{\sqrt[3]{5 x} \sqrt[3]{{(4 y)}^{2}}}{y^{3} \cdot 4}$

Step 5.2

Apply the product rule to $4 y$ .

$\frac{\sqrt[3]{5 x} \sqrt[3]{4^{2} y^{2}}}{y^{3} \cdot 4}$

Step 5.3

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{5 x} \sqrt[3]{16 y^{2}}}{y^{3} \cdot 4}$

Step 5.4

Rewrite $16 y^{2}$ as $2^{3} \cdot (2 y^{2})$ .

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Step 5.4.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{5 x} \sqrt[3]{8 (2) y^{2}}}{y^{3} \cdot 4}$

Step 5.4.2

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[3]{5 x} \sqrt[3]{2^{3} \cdot 2 y^{2}}}{y^{3} \cdot 4}$

Step 5.4.3

Add parentheses.

$\frac{\sqrt[3]{5 x} \sqrt[3]{2^{3} \cdot (2 y^{2})}}{y^{3} \cdot 4}$

Step 5.5

Pull terms out from under the radical.

$\frac{\sqrt[3]{5 x} \cdot 2 \sqrt[3]{2 y^{2}}}{y^{3} \cdot 4}$

Step 5.6

Combine exponents.

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Step 5.6.1

Combine using the product rule for radicals.

$\frac{2 \sqrt[3]{5 x (2 y^{2})}}{y^{3} \cdot 4}$

Step 5.6.2

Multiply $2$ by $5$ .

$\frac{2 \sqrt[3]{10 x y^{2}}}{y^{3} \cdot 4}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $2$ and $4$ .

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Step 6.1.1

Factor $2$ out of $2 \sqrt[3]{10 x y^{2}}$ .

$\frac{2 (\sqrt[3]{10 x y^{2}})}{y^{3} \cdot 4}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $y^{3} \cdot 4$ .

$\frac{2 (\sqrt[3]{10 x y^{2}})}{2 (y^{3} \cdot 2)}$

Step 6.1.2.2

Cancel the common factor.

$\frac{2 \sqrt[3]{10 x y^{2}}}{2 (y^{3} \cdot 2)}$

Step 6.1.2.3

Rewrite the expression.

$\frac{\sqrt[3]{10 x y^{2}}}{y^{3} \cdot 2}$

Step 6.2

Move $2$ to the left of $y^{3}$ .

$\frac{\sqrt[3]{10 x y^{2}}}{2 y^{3}}$