Trigonometry Examples

$r = \sqrt[3]{\frac{A^{2} B}{C}}$

Step 1

Rewrite $\sqrt[3]{\frac{A^{2} B}{C}}$ as $\frac{\sqrt[3]{A^{2} B}}{\sqrt[3]{C}}$ .

$r = \frac{\sqrt[3]{A^{2} B}}{\sqrt[3]{C}}$

Step 2

Multiply $\frac{\sqrt[3]{A^{2} B}}{\sqrt[3]{C}}$ by $\frac{{\sqrt[3]{C}}^{2}}{{\sqrt[3]{C}}^{2}}$ .

$r = \frac{\sqrt[3]{A^{2} B}}{\sqrt[3]{C}} \cdot \frac{{\sqrt[3]{C}}^{2}}{{\sqrt[3]{C}}^{2}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[3]{A^{2} B}}{\sqrt[3]{C}}$ by $\frac{{\sqrt[3]{C}}^{2}}{{\sqrt[3]{C}}^{2}}$ .

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{\sqrt[3]{C} {\sqrt[3]{C}}^{2}}$

Step 3.2

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

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{{\sqrt[3]{C}}^{1} {\sqrt[3]{C}}^{2}}$

Step 3.3

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

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{{\sqrt[3]{C}}^{1 + 2}}$

Step 3.4

Add $1$ and $2$ .

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{{\sqrt[3]{C}}^{3}}$

Step 3.5

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

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Step 3.5.1

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

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{{(C^{\frac{1}{3}})}^{3}}$

Step 3.5.2

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

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{C^{\frac{1}{3} \cdot 3}}$

Step 3.5.3

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

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{C^{\frac{3}{3}}}$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{C^{\frac{3}{3}}}$

Step 3.5.4.2

Rewrite the expression.

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{C^{1}}$

Step 3.5.5

Simplify.

$r = \frac{\sqrt[3]{A^{2} B} {\sqrt[3]{C}}^{2}}{C}$

Step 4

Rewrite ${\sqrt[3]{C}}^{2}$ as $\sqrt[3]{C^{2}}$ .

$r = \frac{\sqrt[3]{A^{2} B} \sqrt[3]{C^{2}}}{C}$

Step 5

Combine using the product rule for radicals.

$r = \frac{\sqrt[3]{A^{2} B C^{2}}}{C}$