Trigonometry Examples

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

Step 1

Pull terms out from under the radical.

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

Step 2

Rewrite $64 a^{4} b^{2}$ as ${(4 a)}^{3} (a b^{2})$ .

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Step 2.1

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

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

Step 2.2

Factor out $a^{3}$ .

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

Step 2.3

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

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

Step 2.4

Add parentheses.

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

Step 3

Pull terms out from under the radical.

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

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

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Step 5.1

Move $a$ .

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

Step 5.2

Multiply $a$ by $a$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $2$ .

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

Step 7

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

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