Trigonometry Examples

$\frac{\sqrt{\sqrt[3]{c^{7} d^{2}}}}{\sqrt{c d}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $c^{7} d^{2}$ as ${(c^{2})}^{3} (c d^{2})$ .

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Step 1.1.1

Factor out $c^{6}$ .

$\frac{\sqrt{\sqrt[3]{c^{6} c d^{2}}}}{\sqrt{c d}}$

Step 1.1.2

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

$\frac{\sqrt{\sqrt[3]{{(c^{2})}^{3} c d^{2}}}}{\sqrt{c d}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt{\sqrt[3]{{(c^{2})}^{3} (c d^{2})}}}{\sqrt{c d}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt{c^{2} \sqrt[3]{c d^{2}}}}{\sqrt{c d}}$

Step 1.3

Pull terms out from under the radical.

$\frac{c \sqrt{\sqrt[3]{c d^{2}}}}{\sqrt{c d}}$

Step 1.4

Rewrite $\sqrt{\sqrt[3]{c d^{2}}}$ as $\sqrt[6]{c d^{2}}$ .

$\frac{c \sqrt[6]{c d^{2}}}{\sqrt{c d}}$

Step 2

Multiply $\frac{c \sqrt[6]{c d^{2}}}{\sqrt{c d}}$ by $\frac{\sqrt{c d}}{\sqrt{c d}}$ .

$\frac{c \sqrt[6]{c d^{2}}}{\sqrt{c d}} \cdot \frac{\sqrt{c d}}{\sqrt{c d}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{c \sqrt[6]{c d^{2}}}{\sqrt{c d}}$ by $\frac{\sqrt{c d}}{\sqrt{c d}}$ .

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{\sqrt{c d} \sqrt{c d}}$

Step 3.1.2

Raise $\sqrt{c d}$ to the power of $1$ .

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{\sqrt{c d}}^{1} \sqrt{c d}}$

Step 3.1.3

Raise $\sqrt{c d}$ to the power of $1$ .

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{\sqrt{c d}}^{1} {\sqrt{c d}}^{1}}$

Step 3.1.4

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

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{\sqrt{c d}}^{1 + 1}}$

Step 3.1.5

Add $1$ and $1$ .

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{\sqrt{c d}}^{2}}$

Step 3.1.6

Rewrite ${\sqrt{c d}}^{2}$ as $c d$ .

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Step 3.1.6.1

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

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{({(c d)}^{\frac{1}{2}})}^{2}}$

Step 3.1.6.2

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

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{(c d)}^{\frac{1}{2} \cdot 2}}$

Step 3.1.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{(c d)}^{\frac{2}{2}}}$

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{(c d)}^{\frac{2}{2}}}$

Step 3.1.6.4.2

Rewrite the expression.

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{{(c d)}^{1}}$

Step 3.1.6.5

Simplify.

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{c d}$

Step 3.2

Cancel the common factor of $c$ .

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Step 3.2.1

Cancel the common factor.

$\frac{c \sqrt[6]{c d^{2}} \sqrt{c d}}{c d}$

Step 3.2.2

Rewrite the expression.

$\frac{\sqrt[6]{c d^{2}} \sqrt{c d}}{d}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

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

$\frac{\sqrt[6]{c d^{2}} {(c d)}^{\frac{1}{2}}}{d}$

Step 4.1.2

Rewrite ${(c d)}^{\frac{1}{2}}$ as ${(c d)}^{\frac{3}{6}}$ .

$\frac{\sqrt[6]{c d^{2}} {(c d)}^{\frac{3}{6}}}{d}$

Step 4.1.3

Rewrite ${(c d)}^{\frac{3}{6}}$ as $\sqrt[6]{{(c d)}^{3}}$ .

$\frac{\sqrt[6]{c d^{2}} \sqrt[6]{{(c d)}^{3}}}{d}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{c d^{2} {(c d)}^{3}}}{d}$

Step 4.3

Apply the product rule to $c d$ .

$\frac{\sqrt[6]{c d^{2} c^{3} d^{3}}}{d}$

Step 4.4

Combine exponents.

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Step 4.4.1

Multiply $c$ by $c^{3}$ by adding the exponents.

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Step 4.4.1.1

Move $c^{3}$ .

$\frac{\sqrt[6]{c^{3} c d^{2} d^{3}}}{d}$

Step 4.4.1.2

Multiply $c^{3}$ by $c$ .

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Step 4.4.1.2.1

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

$\frac{\sqrt[6]{c^{3} c^{1} d^{2} d^{3}}}{d}$

Step 4.4.1.2.2

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

$\frac{\sqrt[6]{c^{3 + 1} d^{2} d^{3}}}{d}$

Step 4.4.1.3

Add $3$ and $1$ .

$\frac{\sqrt[6]{c^{4} d^{2} d^{3}}}{d}$

Step 4.4.2

Multiply $d^{2}$ by $d^{3}$ by adding the exponents.

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Step 4.4.2.1

Move $d^{3}$ .

$\frac{\sqrt[6]{c^{4} (d^{3} d^{2})}}{d}$

Step 4.4.2.2

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

$\frac{\sqrt[6]{c^{4} d^{3 + 2}}}{d}$

Step 4.4.2.3

Add $3$ and $2$ .

$\frac{\sqrt[6]{c^{4} d^{5}}}{d}$