Pre-Algebra Examples

$\sqrt[3]{6 a^{3} b^{4}} \cdot \sqrt[3]{18 a^{5} b^{8}}$

Step 1

Rewrite $6 a^{3} b^{4}$ as ${(a b)}^{3} \cdot (6 b)$ .

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Step 1.1

Factor out $b^{3}$ .

$\sqrt[3]{6 a^{3} (b^{3} b)} \cdot \sqrt[3]{18 a^{5} b^{8}}$

Step 1.2

Move $6$ .

$\sqrt[3]{a^{3} b^{3} \cdot 6 b} \cdot \sqrt[3]{18 a^{5} b^{8}}$

Step 1.3

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

$\sqrt[3]{{(a b)}^{3} \cdot 6 b} \cdot \sqrt[3]{18 a^{5} b^{8}}$

Step 1.4

Add parentheses.

$\sqrt[3]{{(a b)}^{3} \cdot (6 b)} \cdot \sqrt[3]{18 a^{5} b^{8}}$

Step 2

Pull terms out from under the radical.

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{18 a^{5} b^{8}}$

Step 3

Rewrite $18 a^{5} b^{8}$ as ${(a b^{2})}^{3} \cdot (18 a^{2} b^{2})$ .

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Step 3.1

Factor out $a^{3}$ .

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{18 (a^{3} a^{2}) b^{8}}$

Step 3.2

Factor out $b^{6}$ .

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{18 (a^{3} a^{2}) (b^{6} b^{2})}$

Step 3.3

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

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{18 (a^{3} a^{2}) ({(b^{2})}^{3} b^{2})}$

Step 3.4

Move $a^{2}$ .

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{18 (a^{3}) {(b^{2})}^{3} a^{2} b^{2}}$

Step 3.5

Move $18$ .

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{a^{3} {(b^{2})}^{3} \cdot 18 a^{2} b^{2}}$

Step 3.6

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

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{{(a b^{2})}^{3} \cdot 18 a^{2} b^{2}}$

Step 3.7

Add parentheses.

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{{(a b^{2})}^{3} \cdot 18 (a^{2} b^{2})}$

Step 3.8

Add parentheses.

$a b \sqrt[3]{6 b} \cdot \sqrt[3]{{(a b^{2})}^{3} \cdot (18 a^{2} b^{2})}$

Step 4

Pull terms out from under the radical.

$a b \sqrt[3]{6 b} \cdot (a b^{2} \sqrt[3]{18 a^{2} b^{2}})$

Step 5

Multiply $a$ by $a$ by adding the exponents.

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Step 5.1

Move $a$ .

$a \cdot a b \sqrt[3]{6 b} \cdot (b^{2} \sqrt[3]{18 a^{2} b^{2}})$

Step 5.2

Multiply $a$ by $a$ .

$a^{2} b \sqrt[3]{6 b} \cdot (b^{2} \sqrt[3]{18 a^{2} b^{2}})$

Step 6

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

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Step 6.1

Move $b^{2}$ .

$a^{2} (b^{2} b) \sqrt[3]{6 b} \cdot \sqrt[3]{18 a^{2} b^{2}}$

Step 6.2

Multiply $b^{2}$ by $b$ .

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Step 6.2.1

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

$a^{2} (b^{2} b^{1}) \sqrt[3]{6 b} \cdot \sqrt[3]{18 a^{2} b^{2}}$

Step 6.2.2

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

$a^{2} b^{2 + 1} \sqrt[3]{6 b} \cdot \sqrt[3]{18 a^{2} b^{2}}$

Step 6.3

Add $2$ and $1$ .

$a^{2} b^{3} \sqrt[3]{6 b} \cdot \sqrt[3]{18 a^{2} b^{2}}$

Step 7

Multiply $a^{2} b^{3} \sqrt[3]{6 b} \sqrt[3]{18 a^{2} b^{2}}$ .

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Step 7.1

Combine using the product rule for radicals.

$a^{2} b^{3} \sqrt[3]{18 a^{2} b^{2} (6 b)}$

Step 7.2

Multiply $6$ by $18$ .

$a^{2} b^{3} \sqrt[3]{108 a^{2} b^{2} b}$

Step 7.3

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

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

Step 7.4

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

$a^{2} b^{3} \sqrt[3]{108 a^{2} b^{1 + 2}}$

Step 7.5

Add $1$ and $2$ .

$a^{2} b^{3} \sqrt[3]{108 a^{2} b^{3}}$

Step 8

Rewrite $108 a^{2} b^{3}$ as ${(3 b)}^{3} \cdot (4 a^{2})$ .

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Step 8.1

Factor $27$ out of $108$ .

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

Step 8.2

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

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

Step 8.3

Move $a^{2}$ .

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

Step 8.4

Move $4$ .

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

Step 8.5

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

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

Step 8.6

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

Rewrite using the commutative property of multiplication.

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

Step 11

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

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Step 11.1

Move $b$ .

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

Step 11.2

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

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.3

Add $1$ and $3$ .

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