Basic Math Examples

$\sqrt[3]{- 9 a^{5} b^{- 11}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt[3]{- 9 a^{5} \frac{1}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 2

Combine exponents.

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Step 2.1

Combine $- 9$ and $\frac{1}{b^{11}}$ .

$\sqrt[3]{a^{5} \frac{- 9}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 2.2

Combine $a^{5}$ and $\frac{- 9}{b^{11}}$ .

$\sqrt[3]{\frac{a^{5} \cdot - 9}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 3

Move $- 9$ to the left of $a^{5}$ .

$\sqrt[3]{\frac{- 9 \cdot a^{5}}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 4

Move the negative in front of the fraction.

$\sqrt[3]{- \frac{9 \cdot a^{5}}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 5

Rewrite $- \frac{9 a^{5}}{b^{11}}$ as ${(- \frac{a}{b^{3}})}^{3} \frac{9 a^{2}}{b^{2}}$ .

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Step 5.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{9 a^{5}}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 5.2

Factor the perfect power $a^{3}$ out of $9 a^{5}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{a^{3} \cdot (9 a^{2})}{b^{11}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 5.3

Factor the perfect power ${(b^{3})}^{3}$ out of $b^{11}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{a^{3} \cdot (9 a^{2})}{{(b^{3})}^{3} b^{2}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 5.4

Rearrange the fraction $\frac{a^{3} \cdot (9 a^{2})}{{(b^{3})}^{3} b^{2}}$ .

$\sqrt[3]{{(- 1)}^{3} ({(\frac{a}{b^{3}})}^{3} \frac{9 a^{2}}{b^{2}})} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 5.5

Rewrite ${(- 1)}^{3} {(\frac{a}{b^{3}})}^{3}$ as ${(- \frac{a}{b^{3}})}^{3}$ .

$\sqrt[3]{{(- \frac{a}{b^{3}})}^{3} \frac{9 a^{2}}{b^{2}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 6

Pull terms out from under the radical.

$- \frac{a}{b^{3}} \sqrt[3]{\frac{9 a^{2}}{b^{2}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 7

Rewrite $\sqrt[3]{\frac{9 a^{2}}{b^{2}}}$ as $\frac{\sqrt[3]{9 a^{2}}}{\sqrt[3]{b^{2}}}$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}}}{\sqrt[3]{b^{2}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 8

Multiply $\frac{\sqrt[3]{9 a^{2}}}{\sqrt[3]{b^{2}}}$ by $\frac{{\sqrt[3]{b^{2}}}^{2}}{{\sqrt[3]{b^{2}}}^{2}}$ .

$- \frac{a}{b^{3}} (\frac{\sqrt[3]{9 a^{2}}}{\sqrt[3]{b^{2}}} \cdot \frac{{\sqrt[3]{b^{2}}}^{2}}{{\sqrt[3]{b^{2}}}^{2}}) \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt[3]{9 a^{2}}}{\sqrt[3]{b^{2}}}$ by $\frac{{\sqrt[3]{b^{2}}}^{2}}{{\sqrt[3]{b^{2}}}^{2}}$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{\sqrt[3]{b^{2}} {\sqrt[3]{b^{2}}}^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.2

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

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{{\sqrt[3]{b^{2}}}^{1} {\sqrt[3]{b^{2}}}^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.3

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

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{{\sqrt[3]{b^{2}}}^{1 + 2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.4

Add $1$ and $2$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{{\sqrt[3]{b^{2}}}^{3}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5

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

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Step 9.5.1

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

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{{(b^{\frac{2}{3}})}^{3}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.2

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

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{2}{3} \cdot 3}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.3

Combine $\frac{2}{3}$ and $3$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{2 \cdot 3}{3}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.4

Multiply $2$ by $3$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{6}{3}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.5

Cancel the common factor of $6$ and $3$ .

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Step 9.5.5.1

Factor $3$ out of $6$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{3 \cdot 2}{3}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.5.2

Cancel the common factors.

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Step 9.5.5.2.1

Factor $3$ out of $3$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{3 \cdot 2}{3 (1)}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.5.2.2

Cancel the common factor.

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{3 \cdot 2}{3 \cdot 1}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.5.2.3

Rewrite the expression.

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{2}{1}}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 9.5.5.2.4

Divide $2$ by $1$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} {\sqrt[3]{b^{2}}}^{2}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

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

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} \sqrt[3]{{(b^{2})}^{2}}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 10.2

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

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Step 10.2.1

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

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} \sqrt[3]{b^{2 \cdot 2}}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 10.2.2

Multiply $2$ by $2$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} \sqrt[3]{b^{4}}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 10.3

Factor out $b^{3}$ .

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} \sqrt[3]{b^{3} b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 10.4

Pull terms out from under the radical.

$- \frac{a}{b^{3}} \cdot \frac{\sqrt[3]{9 a^{2}} b \sqrt[3]{b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 10.5

Combine using the product rule for radicals.

$- \frac{a}{b^{3}} \cdot \frac{b \sqrt[3]{9 a^{2} b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 11

Simplify terms.

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Step 11.1

Cancel the common factor of $b$ .

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Step 11.1.1

Move the leading negative in $- \frac{a}{b^{3}}$ into the numerator.

$\frac{- a}{b^{3}} \cdot \frac{b \sqrt[3]{9 a^{2} b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 11.1.2

Factor $b$ out of $b^{3}$ .

$\frac{- a}{b \cdot b^{2}} \cdot \frac{b \sqrt[3]{9 a^{2} b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 11.1.3

Factor $b$ out of $b \sqrt[3]{9 a^{2} b}$ .

$\frac{- a}{b \cdot b^{2}} \cdot \frac{b (\sqrt[3]{9 a^{2} b})}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 11.1.4

Cancel the common factor.

$\frac{- a}{b \cdot b^{2}} \cdot \frac{b \sqrt[3]{9 a^{2} b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 11.1.5

Rewrite the expression.

$\frac{- a}{b^{2}} \cdot \frac{\sqrt[3]{9 a^{2} b}}{b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 11.2

Multiply $\frac{- a}{b^{2}}$ by $\frac{\sqrt[3]{9 a^{2} b}}{b^{2}}$ .

$\frac{- a \sqrt[3]{9 a^{2} b}}{b^{2} b^{2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 12

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

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Step 12.1

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

$\frac{- a \sqrt[3]{9 a^{2} b}}{b^{2 + 2}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 12.2

Add $2$ and $2$ .

$\frac{- a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 13

Move the negative in front of the fraction.

$- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot \sqrt[3]{3 a^{4} b^{2}}$

Step 14

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

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Step 14.1

Factor out $a^{3}$ .

$- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot \sqrt[3]{3 (a^{3} a) b^{2}}$

Step 14.2

Reorder $3$ and $a^{3}$ .

$- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot \sqrt[3]{a^{3} \cdot 3 a b^{2}}$

Step 14.3

Add parentheses.

$- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot \sqrt[3]{a^{3} \cdot 3 (a b^{2})}$

Step 14.4

Add parentheses.

$- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot \sqrt[3]{a^{3} \cdot (3 a b^{2})}$

Step 15

Pull terms out from under the radical.

$- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} \cdot (a \sqrt[3]{3 a b^{2}})$

Step 16

Multiply $- \frac{a \sqrt[3]{9 a^{2} b}}{b^{4}} (a \sqrt[3]{3 a b^{2}})$ .

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Step 16.1

Combine $a$ and $\frac{a \sqrt[3]{9 a^{2} b}}{b^{4}}$ .

$- \frac{a (a \sqrt[3]{9 a^{2} b})}{b^{4}} \sqrt[3]{3 a b^{2}}$

Step 16.2

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

$- \frac{a^{1} a \sqrt[3]{9 a^{2} b}}{b^{4}} \sqrt[3]{3 a b^{2}}$

Step 16.3

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

$- \frac{a^{1} a^{1} \sqrt[3]{9 a^{2} b}}{b^{4}} \sqrt[3]{3 a b^{2}}$

Step 16.4

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

$- \frac{a^{1 + 1} \sqrt[3]{9 a^{2} b}}{b^{4}} \sqrt[3]{3 a b^{2}}$

Step 16.5

Add $1$ and $1$ .

$- \frac{a^{2} \sqrt[3]{9 a^{2} b}}{b^{4}} \sqrt[3]{3 a b^{2}}$

Step 16.6

Combine $\sqrt[3]{3 a b^{2}}$ and $\frac{a^{2} \sqrt[3]{9 a^{2} b}}{b^{4}}$ .

$- \frac{\sqrt[3]{3 a b^{2}} (a^{2} \sqrt[3]{9 a^{2} b})}{b^{4}}$

Step 16.7

Combine using the product rule for radicals.

$- \frac{a^{2} \sqrt[3]{3 a b^{2} (9 a^{2} b)}}{b^{4}}$

Step 16.8

Multiply $9$ by $3$ .

$- \frac{a^{2} \sqrt[3]{27 a b^{2} (a^{2} b)}}{b^{4}}$

Step 16.9

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

$- \frac{a^{2} \sqrt[3]{27 (a^{2} a^{1}) b^{2} b}}{b^{4}}$

Step 16.10

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

$- \frac{a^{2} \sqrt[3]{27 a^{2 + 1} b^{2} b}}{b^{4}}$

Step 16.11

Add $2$ and $1$ .

$- \frac{a^{2} \sqrt[3]{27 a^{3} b^{2} b}}{b^{4}}$

Step 16.12

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

$- \frac{a^{2} \sqrt[3]{27 a^{3} (b^{1} b^{2})}}{b^{4}}$

Step 16.13

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

$- \frac{a^{2} \sqrt[3]{27 a^{3} b^{1 + 2}}}{b^{4}}$

Step 16.14

Add $1$ and $2$ .

$- \frac{a^{2} \sqrt[3]{27 a^{3} b^{3}}}{b^{4}}$

Step 17

Simplify the numerator.

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Step 17.1

Rewrite $27 a^{3} b^{3}$ as ${(3 a b)}^{3}$ .

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

Step 17.2

Pull terms out from under the radical, assuming real numbers.

$- \frac{a^{2} \cdot 3 a b}{b^{4}}$

Step 17.3

Combine exponents.

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Step 17.3.1

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

$- \frac{3 (a^{2} a^{1}) b}{b^{4}}$

Step 17.3.2

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

$- \frac{3 a^{2 + 1} b}{b^{4}}$

Step 17.3.3

Add $2$ and $1$ .

$- \frac{3 a^{3} b}{b^{4}}$

Step 18

Cancel the common factor of $b$ and $b^{4}$ .

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Step 18.1

Factor $b$ out of $3 a^{3} b$ .

$- \frac{b (3 a^{3})}{b^{4}}$

Step 18.2

Cancel the common factors.

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Step 18.2.1

Factor $b$ out of $b^{4}$ .

$- \frac{b (3 a^{3})}{b \cdot b^{3}}$

Step 18.2.2

Cancel the common factor.

$- \frac{b (3 a^{3})}{b \cdot b^{3}}$

Step 18.2.3

Rewrite the expression.

$- \frac{3 a^{3}}{b^{3}}$