Trigonometry Examples

$\frac{\sqrt[4]{b^{9} c^{3}}}{\sqrt{b c}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $b^{9} c^{3}$ as ${(b^{2})}^{4} (b c^{3})$ .

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Step 1.1.1

Factor out $b^{8}$ .

$\frac{\sqrt[4]{b^{8} b c^{3}}}{\sqrt{b c}}$

Step 1.1.2

Rewrite $b^{8}$ as ${(b^{2})}^{4}$ .

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{b^{2} \sqrt[4]{b c^{3}}}{\sqrt{b c}}$ by $\frac{\sqrt{b c}}{\sqrt{b c}}$ .

$\frac{b^{2} \sqrt[4]{b c^{3}}}{\sqrt{b c}} \cdot \frac{\sqrt{b c}}{\sqrt{b c}}$

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 $\frac{b^{2} \sqrt[4]{b c^{3}}}{\sqrt{b c}}$ by $\frac{\sqrt{b c}}{\sqrt{b c}}$ .

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

Step 3.1.2

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

$\frac{b^{2} \sqrt[4]{b c^{3}} \sqrt{b c}}{{\sqrt{b c}}^{1} \sqrt{b c}}$

Step 3.1.3

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

$\frac{b^{2} \sqrt[4]{b c^{3}} \sqrt{b c}}{{\sqrt{b c}}^{1} {\sqrt{b c}}^{1}}$

Step 3.1.4

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

$\frac{b^{2} \sqrt[4]{b c^{3}} \sqrt{b c}}{{\sqrt{b c}}^{1 + 1}}$

Step 3.1.5

Add $1$ and $1$ .

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

Step 3.1.6

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

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Step 3.1.6.1

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

$\frac{b^{2} \sqrt[4]{b c^{3}} \sqrt{b c}}{{({(b c)}^{\frac{1}{2}})}^{2}}$

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

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.

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

Step 3.1.6.4.2

Rewrite the expression.

$\frac{b^{2} \sqrt[4]{b c^{3}} \sqrt{b c}}{{(b c)}^{1}}$

Step 3.1.6.5

Simplify.

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

Step 3.2

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

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Step 3.2.1

Factor $b$ out of $b^{2} \sqrt[4]{b c^{3}} \sqrt{b c}$ .

$\frac{b (b \sqrt[4]{b c^{3}} \sqrt{b c})}{b c}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $b$ out of $b c$ .

$\frac{b (b \sqrt[4]{b c^{3}} \sqrt{b c})}{b (c)}$

Step 3.2.2.2

Cancel the common factor.

$\frac{b (b \sqrt[4]{b c^{3}} \sqrt{b c})}{b c}$

Step 3.2.2.3

Rewrite the expression.

$\frac{b \sqrt[4]{b c^{3}} \sqrt{b c}}{c}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $4$ .

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Step 4.1.1

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

$\frac{b ({(b c)}^{\frac{1}{2}} \sqrt[4]{b c^{3}})}{c}$

Step 4.1.2

Rewrite ${(b c)}^{\frac{1}{2}}$ as ${(b c)}^{\frac{2}{4}}$ .

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

Step 4.1.3

Rewrite ${(b c)}^{\frac{2}{4}}$ as $\sqrt[4]{{(b c)}^{2}}$ .

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Apply the product rule to $b c$ .

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

Step 4.4

Combine exponents.

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Step 4.4.1

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

$\frac{b \sqrt[4]{b^{1} b^{2} c^{2} c^{3}}}{c}$

Step 4.4.2

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

$\frac{b \sqrt[4]{b^{1 + 2} c^{2} c^{3}}}{c}$

Step 4.4.3

Add $1$ and $2$ .

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

Step 4.4.4

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

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Step 4.4.4.1

Move $c^{3}$ .

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

Step 4.4.4.2

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

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

Step 4.4.4.3

Add $3$ and $2$ .

$\frac{b \sqrt[4]{b^{3} c^{5}}}{c}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $b^{3} c^{5}$ as $c^{4} (b^{3} c)$ .

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Step 5.1.1

Factor out $c^{4}$ .

$\frac{b \sqrt[4]{b^{3} (c^{4} c)}}{c}$

Step 5.1.2

Reorder $b^{3}$ and $c^{4}$ .

$\frac{b \sqrt[4]{c^{4} b^{3} c}}{c}$

Step 5.1.3

Add parentheses.

$\frac{b \sqrt[4]{c^{4} (b^{3} c)}}{c}$

Step 5.2

Pull terms out from under the radical.

$\frac{b c \sqrt[4]{b^{3} c}}{c}$

Step 6

Cancel the common factor of $c$ .

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Step 6.1

Cancel the common factor.

$\frac{b c \sqrt[4]{b^{3} c}}{c}$

Step 6.2

Divide $b \sqrt[4]{b^{3} c}$ by $1$ .

$b \sqrt[4]{b^{3} c}$