Trigonometry Examples

$\sqrt[3]{\frac{12 (x^{2})}{16 y}}$

Step 1

Reduce the expression $\frac{12 (x^{2})}{16 y}$ by cancelling the common factors.

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Step 1.1

Factor $4$ out of $12 (x^{2})$ .

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

Step 1.2

Factor $4$ out of $16 y$ .

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

Step 1.3

Cancel the common factor.

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

Step 1.4

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $2$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

Apply the product rule to $4 y$ .

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

Step 5.3

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

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

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 $16$ .

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

Step 5.4.2

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

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

Step 5.4.3

Add parentheses.

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

Step 5.5

Pull terms out from under the radical.

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

Step 5.6

Combine exponents.

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Step 5.6.1

Combine using the product rule for radicals.

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

Step 5.6.2

Multiply $2$ by $3$ .

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

Step 6

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

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Step 6.1

Factor $2$ out of $2 \sqrt[3]{6 x^{2} y^{2}}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $4 y$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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