Precalculus Examples

$\sqrt[3]{a^{2} b} \sqrt[3]{64 a^{4} b}$

Step 1

Rewrite $64 a^{4} b$ as ${(4 a)}^{3} (a b)$ .

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Step 1.1

Rewrite $64$ as $4^{3}$ .

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

Step 1.2

Factor out $a^{3}$ .

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

Step 1.3

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

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

Step 1.4

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

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

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $2$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Add $1$ and $1$ .

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

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Step 6.1

Move $a$ .

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

Step 6.2

Multiply $a$ by $a$ .

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

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