Basic Math Examples

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

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

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Step 2.1

Move $b$ .

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

Step 2.2

Multiply $b$ by $b$ .

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

Step 3

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

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Step 3.1

Combine using the product rule for radicals.

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

Step 3.2

Multiply $2$ by $4$ .

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

Step 3.3

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

$2 a b^{2} \sqrt[3]{8 a^{2 + 2} b \cdot b^{2}}$

Step 3.4

Add $2$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

Move $b^{2}$ .

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

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

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

Step 3.5.3

Add $2$ and $1$ .

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

Step 4

Rewrite $8 a^{4} b^{3}$ as ${(2 a b)}^{3} a$ .

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Step 4.1

Rewrite $8$ as $2^{3}$ .

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

Step 4.2

Factor out $a^{3}$ .

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

Step 4.3

Move $a$ .

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

Step 4.4

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

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Step 6.1

Move $a$ .

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

Step 6.2

Multiply $a$ by $a$ .

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

Step 7

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

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Step 7.1

Move $b$ .

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

Add $1$ and $2$ .

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

Step 8

Multiply $2$ by $2$ .

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