Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

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

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

Step 2.1.2

Raise $\sqrt[3]{4 x y^{2}}$ to the power of $1$ .

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

Step 2.1.3

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

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

Step 2.1.4

Add $1$ and $2$ .

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

Step 2.1.5

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

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x y^{2}}$ as ${(4 x y^{2})}^{\frac{1}{3}}$ .

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Cancel the common factor of $3$ .

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Step 2.1.5.4.1

Cancel the common factor.

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

Step 2.1.5.4.2

Rewrite the expression.

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

Step 2.1.5.5

Simplify.

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

Step 2.2

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

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Step 2.2.1

Factor $2$ out of $2 x {\sqrt[3]{4 x y^{2}}}^{2}$ .

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $2$ out of $4 x y^{2}$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.3

Cancel the common factor of $x$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $4 x y^{2}$ .

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

Step 3.3

Apply the product rule to $4 x$ .

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

Step 3.4

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

$\frac{\sqrt[3]{16 x^{2} {(y^{2})}^{2}}}{2 y^{2}}$

Step 3.5

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 3.5.1

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

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

Step 3.5.2

Multiply $2$ by $2$ .

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

Step 3.6

Rewrite $16 x^{2} y^{4}$ as ${(2 y)}^{3} \cdot (2 x^{2} y)$ .

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Step 3.6.1

Factor $8$ out of $16$ .

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

Step 3.6.2

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

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

Step 3.6.3

Factor out $y^{3}$ .

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

Step 3.6.4

Move $x^{2}$ .

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

Step 3.6.5

Move $2$ .

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

Step 3.6.6

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

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

Step 3.6.7

Add parentheses.

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

Step 3.6.8

Add parentheses.

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

Step 3.7

Pull terms out from under the radical.

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $y$ and $y^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $y$ out of $y^{2}$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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