Trigonometry Examples

$\frac{3 a}{\sqrt[5]{8 a^{7} b^{13}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8 a^{7} b^{13}$ as ${(a b^{2})}^{5} \cdot (8 a^{2} b^{3})$ .

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Step 1.1.1

Factor out $a^{5}$ .

$\frac{3 a}{\sqrt[5]{8 (a^{5} a^{2}) b^{13}}}$

Step 1.1.2

Factor out $b^{10}$ .

$\frac{3 a}{\sqrt[5]{8 (a^{5} a^{2}) (b^{10} b^{3})}}$

Step 1.1.3

Rewrite $b^{10}$ as ${(b^{2})}^{5}$ .

$\frac{3 a}{\sqrt[5]{8 (a^{5} a^{2}) ({(b^{2})}^{5} b^{3})}}$

Step 1.1.4

Move $a^{2}$ .

$\frac{3 a}{\sqrt[5]{8 (a^{5}) {(b^{2})}^{5} a^{2} b^{3}}}$

Step 1.1.5

Move $8$ .

$\frac{3 a}{\sqrt[5]{a^{5} {(b^{2})}^{5} \cdot 8 a^{2} b^{3}}}$

Step 1.1.6

Rewrite $(a^{5}) {(b^{2})}^{5}$ as ${(a b^{2})}^{5}$ .

$\frac{3 a}{\sqrt[5]{{(a b^{2})}^{5} \cdot 8 a^{2} b^{3}}}$

Step 1.1.7

Add parentheses.

$\frac{3 a}{\sqrt[5]{{(a b^{2})}^{5} \cdot 8 (a^{2} b^{3})}}$

Step 1.1.8

Add parentheses.

$\frac{3 a}{\sqrt[5]{{(a b^{2})}^{5} \cdot (8 a^{2} b^{3})}}$

Step 1.2

Pull terms out from under the radical.

$\frac{3 a}{a b^{2} \sqrt[5]{8 a^{2} b^{3}}}$

Step 2

Cancel the common factor of $a$ .

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Step 2.1

Cancel the common factor.

$\frac{3 a}{a b^{2} \sqrt[5]{8 a^{2} b^{3}}}$

Step 2.2

Rewrite the expression.

$\frac{3}{b^{2} \sqrt[5]{8 a^{2} b^{3}}}$

Step 3

Multiply $\frac{3}{b^{2} \sqrt[5]{8 a^{2} b^{3}}}$ by $\frac{{\sqrt[5]{8 a^{2} b^{3}}}^{4}}{{\sqrt[5]{8 a^{2} b^{3}}}^{4}}$ .

$\frac{3}{b^{2} \sqrt[5]{8 a^{2} b^{3}}} \cdot \frac{{\sqrt[5]{8 a^{2} b^{3}}}^{4}}{{\sqrt[5]{8 a^{2} b^{3}}}^{4}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3}{b^{2} \sqrt[5]{8 a^{2} b^{3}}}$ by $\frac{{\sqrt[5]{8 a^{2} b^{3}}}^{4}}{{\sqrt[5]{8 a^{2} b^{3}}}^{4}}$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} \sqrt[5]{8 a^{2} b^{3}} {\sqrt[5]{8 a^{2} b^{3}}}^{4}}$

Step 4.2

Move $\sqrt[5]{8 a^{2} b^{3}}$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} (\sqrt[5]{8 a^{2} b^{3}} {\sqrt[5]{8 a^{2} b^{3}}}^{4})}$

Step 4.3

Raise $\sqrt[5]{8 a^{2} b^{3}}$ to the power of $1$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} ({\sqrt[5]{8 a^{2} b^{3}}}^{1} {\sqrt[5]{8 a^{2} b^{3}}}^{4})}$

Step 4.4

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

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {\sqrt[5]{8 a^{2} b^{3}}}^{1 + 4}}$

Step 4.5

Add $1$ and $4$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {\sqrt[5]{8 a^{2} b^{3}}}^{5}}$

Step 4.6

Rewrite ${\sqrt[5]{8 a^{2} b^{3}}}^{5}$ as $8 a^{2} b^{3}$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8 a^{2} b^{3}}$ as ${(8 a^{2} b^{3})}^{\frac{1}{5}}$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {({(8 a^{2} b^{3})}^{\frac{1}{5}})}^{5}}$

Step 4.6.2

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

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {(8 a^{2} b^{3})}^{\frac{1}{5} \cdot 5}}$

Step 4.6.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {(8 a^{2} b^{3})}^{\frac{5}{5}}}$

Step 4.6.4

Cancel the common factor of $5$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {(8 a^{2} b^{3})}^{\frac{5}{5}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} {(8 a^{2} b^{3})}^{1}}$

Step 4.6.5

Simplify.

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{2} (8 a^{2} b^{3})}$

Step 5

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

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Step 5.1

Move $b^{3}$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{3} b^{2} (8 a^{2})}$

Step 5.2

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

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{3 + 2} (8 a^{2})}$

Step 5.3

Add $3$ and $2$ .

$\frac{3 {\sqrt[5]{8 a^{2} b^{3}}}^{4}}{b^{5} (8 a^{2})}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[5]{8 a^{2} b^{3}}}^{4}$ as $\sqrt[5]{{(8 a^{2} b^{3})}^{4}}$ .

$\frac{3 \sqrt[5]{{(8 a^{2} b^{3})}^{4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.2

Apply the product rule to $8 a^{2} b^{3}$ .

$\frac{3 \sqrt[5]{{(8 a^{2})}^{4} {(b^{3})}^{4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.3

Apply the product rule to $8 a^{2}$ .

$\frac{3 \sqrt[5]{8^{4} {(a^{2})}^{4} {(b^{3})}^{4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.4

Raise $8$ to the power of $4$ .

$\frac{3 \sqrt[5]{4096 {(a^{2})}^{4} {(b^{3})}^{4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.5

Multiply the exponents in ${(a^{2})}^{4}$ .

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Step 6.5.1

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

$\frac{3 \sqrt[5]{4096 a^{2 \cdot 4} {(b^{3})}^{4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.5.2

Multiply $2$ by $4$ .

$\frac{3 \sqrt[5]{4096 a^{8} {(b^{3})}^{4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.6

Multiply the exponents in ${(b^{3})}^{4}$ .

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Step 6.6.1

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

$\frac{3 \sqrt[5]{4096 a^{8} b^{3 \cdot 4}}}{b^{5} \cdot 8 a^{2}}$

Step 6.6.2

Multiply $3$ by $4$ .

$\frac{3 \sqrt[5]{4096 a^{8} b^{12}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7

Rewrite $4096 a^{8} b^{12}$ as ${(4 a b^{2})}^{5} \cdot (4 a^{3} b^{2})$ .

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Step 6.7.1

Factor $1024$ out of $4096$ .

$\frac{3 \sqrt[5]{1024 (4) a^{8} b^{12}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.2

Rewrite $1024$ as $4^{5}$ .

$\frac{3 \sqrt[5]{4^{5} \cdot 4 a^{8} b^{12}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.3

Factor out $a^{5}$ .

$\frac{3 \sqrt[5]{4^{5} \cdot 4 (a^{5} a^{3}) b^{12}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.4

Factor out $b^{10}$ .

$\frac{3 \sqrt[5]{4^{5} \cdot 4 (a^{5} a^{3}) (b^{10} b^{2})}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.5

Rewrite $b^{10}$ as ${(b^{2})}^{5}$ .

$\frac{3 \sqrt[5]{4^{5} \cdot 4 (a^{5} a^{3}) ({(b^{2})}^{5} b^{2})}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.6

Move $a^{3}$ .

$\frac{3 \sqrt[5]{4^{5} \cdot 4 (a^{5}) {(b^{2})}^{5} a^{3} b^{2}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.7

Move $4$ .

$\frac{3 \sqrt[5]{(4^{5} (a^{5})) {(b^{2})}^{5} \cdot 4 a^{3} b^{2}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.8

Rewrite $(4^{5} (a^{5})) {(b^{2})}^{5}$ as ${(4 a b^{2})}^{5}$ .

$\frac{3 \sqrt[5]{{(4 a b^{2})}^{5} \cdot 4 a^{3} b^{2}}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.9

Add parentheses.

$\frac{3 \sqrt[5]{{(4 a b^{2})}^{5} \cdot 4 (a^{3} b^{2})}}{b^{5} \cdot 8 a^{2}}$

Step 6.7.10

Add parentheses.

$\frac{3 \sqrt[5]{{(4 a b^{2})}^{5} \cdot (4 a^{3} b^{2})}}{b^{5} \cdot 8 a^{2}}$

Step 6.8

Pull terms out from under the radical.

$\frac{3 \cdot 4 a b^{2} \sqrt[5]{4 a^{3} b^{2}}}{b^{5} \cdot 8 a^{2}}$

Step 6.9

Multiply $3$ by $4$ .

$\frac{12 a b^{2} \sqrt[5]{4 a^{3} b^{2}}}{b^{5} \cdot 8 a^{2}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $12$ and $8$ .

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Step 7.1.1

Factor $4$ out of $12 a b^{2} \sqrt[5]{4 a^{3} b^{2}}$ .

$\frac{4 (3 a b^{2} \sqrt[5]{4 a^{3} b^{2}})}{b^{5} \cdot 8 a^{2}}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $4$ out of $b^{5} \cdot 8 a^{2}$ .

$\frac{4 (3 a b^{2} \sqrt[5]{4 a^{3} b^{2}})}{4 (b^{5} \cdot 2 a^{2})}$

Step 7.1.2.2

Cancel the common factor.

$\frac{4 (3 a b^{2} \sqrt[5]{4 a^{3} b^{2}})}{4 (b^{5} \cdot 2 a^{2})}$

Step 7.1.2.3

Rewrite the expression.

$\frac{3 a b^{2} \sqrt[5]{4 a^{3} b^{2}}}{b^{5} \cdot 2 a^{2}}$

Step 7.2

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

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Step 7.2.1

Factor $a$ out of $3 a b^{2} \sqrt[5]{4 a^{3} b^{2}}$ .

$\frac{a (3 b^{2} \sqrt[5]{4 a^{3} b^{2}})}{b^{5} \cdot 2 a^{2}}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $a$ out of $b^{5} \cdot 2 a^{2}$ .

$\frac{a (3 b^{2} \sqrt[5]{4 a^{3} b^{2}})}{a (b^{5} \cdot 2 a)}$

Step 7.2.2.2

Cancel the common factor.

$\frac{a (3 b^{2} \sqrt[5]{4 a^{3} b^{2}})}{a (b^{5} \cdot 2 a)}$

Step 7.2.2.3

Rewrite the expression.

$\frac{3 b^{2} \sqrt[5]{4 a^{3} b^{2}}}{b^{5} \cdot 2 a}$

Step 7.3

Cancel the common factor of $b^{2}$ and $b^{5}$ .

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Step 7.3.1

Factor $b^{2}$ out of $3 b^{2} \sqrt[5]{4 a^{3} b^{2}}$ .

$\frac{b^{2} (3 \sqrt[5]{4 a^{3} b^{2}})}{b^{5} \cdot 2 a}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $b^{2}$ out of $b^{5} \cdot 2 a$ .

$\frac{b^{2} (3 \sqrt[5]{4 a^{3} b^{2}})}{b^{2} (b^{3} \cdot 2 a)}$

Step 7.3.2.2

Cancel the common factor.

$\frac{b^{2} (3 \sqrt[5]{4 a^{3} b^{2}})}{b^{2} (b^{3} \cdot 2 a)}$

Step 7.3.2.3

Rewrite the expression.

$\frac{3 \sqrt[5]{4 a^{3} b^{2}}}{b^{3} \cdot 2 a}$

Step 7.4

Move $2$ to the left of $b^{3}$ .

$\frac{3 \sqrt[5]{4 a^{3} b^{2}}}{2 b^{3} a}$