Algebra Examples

$\sqrt[3]{- 3 a^{7} b^{4}} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1

Rewrite $- 3 a^{7} b^{4}$ as ${(- 1 a^{2} b)}^{3} \cdot (3 a b)$ .

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Step 1.1

Rewrite $- 3$ as $- 1 (3)$ .

$\sqrt[3]{- 1 (3) a^{7} b^{4}} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.2

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

$\sqrt[3]{{(- 1)}^{3} \cdot 3 a^{7} b^{4}} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.3

Factor out $a^{6}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 3 (a^{6} a) b^{4}} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.4

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

$\sqrt[3]{{(- 1)}^{3} \cdot 3 ({(a^{2})}^{3} a) b^{4}} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.5

Factor out $b^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 3 ({(a^{2})}^{3} a) (b^{3} b)} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.6

Move $a$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 3 ({(a^{2})}^{3}) b^{3} a b} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.7

Move $3$ .

$\sqrt[3]{({(- 1)}^{3} ({(a^{2})}^{3})) b^{3} \cdot 3 a b} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.8

Rewrite $({(- 1)}^{3} ({(a^{2})}^{3})) b^{3}$ as ${(- 1 a^{2} b)}^{3}$ .

$\sqrt[3]{{(- 1 a^{2} b)}^{3} \cdot 3 a b} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.9

Add parentheses.

$\sqrt[3]{{(- 1 a^{2} b)}^{3} \cdot 3 (a b)} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 1.10

Add parentheses.

$\sqrt[3]{{(- 1 a^{2} b)}^{3} \cdot (3 a b)} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 2

Pull terms out from under the radical.

$- 1 a^{2} b \sqrt[3]{3 a b} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 3

Rewrite $- 1 a^{2}$ as $- a^{2}$ .

$- a^{2} b \sqrt[3]{3 a b} \cdot \sqrt[3]{36 a^{6} b^{2}}$

Step 4

Rewrite $36 a^{6} b^{2}$ as ${(a^{2})}^{3} \cdot (36 b^{2})$ .

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Step 4.1

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

$- a^{2} b \sqrt[3]{3 a b} \cdot \sqrt[3]{36 {(a^{2})}^{3} b^{2}}$

Step 4.2

Reorder $36$ and ${(a^{2})}^{3}$ .

$- a^{2} b \sqrt[3]{3 a b} \cdot \sqrt[3]{{(a^{2})}^{3} \cdot 36 b^{2}}$

Step 4.3

Add parentheses.

$- a^{2} b \sqrt[3]{3 a b} \cdot \sqrt[3]{{(a^{2})}^{3} \cdot (36 b^{2})}$

Step 5

Pull terms out from under the radical.

$- a^{2} b \sqrt[3]{3 a b} \cdot (a^{2} \sqrt[3]{36 b^{2}})$

Step 6

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

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Step 6.1

Move $a^{2}$ .

$- (a^{2} a^{2}) b \sqrt[3]{3 a b} \cdot \sqrt[3]{36 b^{2}}$

Step 6.2

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

$- a^{2 + 2} b \sqrt[3]{3 a b} \cdot \sqrt[3]{36 b^{2}}$

Step 6.3

Add $2$ and $2$ .

$- a^{4} b \sqrt[3]{3 a b} \cdot \sqrt[3]{36 b^{2}}$

Step 7

Multiply $- a^{4} b \sqrt[3]{3 a b} \sqrt[3]{36 b^{2}}$ .

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Step 7.1

Combine using the product rule for radicals.

$- a^{4} b \sqrt[3]{36 b^{2} (3 a b)}$

Step 7.2

Multiply $3$ by $36$ .

$- a^{4} b \sqrt[3]{108 b^{2} (a b)}$

Step 7.3

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

$- a^{4} b \sqrt[3]{108 (a (b^{2} b^{1}))}$

Step 7.4

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

$- a^{4} b \sqrt[3]{108 (a b^{2 + 1})}$

Step 7.5

Add $2$ and $1$ .

$- a^{4} b \sqrt[3]{108 (a b^{3})}$

Step 8

Rewrite $108 a b^{3}$ as ${(3 b)}^{3} \cdot (4 a)$ .

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Step 8.1

Factor $27$ out of $108$ .

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

Step 8.2

Rewrite $27$ as $3^{3}$ .

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

Step 8.3

Move $a$ .

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

Step 8.4

Move $4$ .

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

Step 8.5

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

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

Step 8.6

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

Multiply $b$ by $b$ by adding the exponents.

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Step 10.1

Move $b$ .

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

Step 10.2

Multiply $b$ by $b$ .

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

Step 11

Multiply $3$ by $- 1$ .

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