Algebra Examples

$(\sqrt[3]{20 b^{7}}) (\sqrt[3]{162 b^{3}})$

Step 1

Rewrite $20 b^{7}$ as ${(b^{2})}^{3} \cdot (20 b)$ .

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Step 1.1

Factor out $b^{6}$ .

$\sqrt[3]{20 (b^{6} b)} \sqrt[3]{162 b^{3}}$

Step 1.2

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

$\sqrt[3]{20 ({(b^{2})}^{3} b)} \sqrt[3]{162 b^{3}}$

Step 1.3

Reorder $20$ and ${(b^{2})}^{3}$ .

$\sqrt[3]{{(b^{2})}^{3} \cdot 20 b} \sqrt[3]{162 b^{3}}$

Step 1.4

Add parentheses.

$\sqrt[3]{{(b^{2})}^{3} \cdot (20 b)} \sqrt[3]{162 b^{3}}$

Step 2

Pull terms out from under the radical.

$b^{2} \sqrt[3]{20 b} \sqrt[3]{162 b^{3}}$

Step 3

Rewrite $162 b^{3}$ as ${(3 b)}^{3} \cdot 6$ .

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Step 3.1

Factor $27$ out of $162$ .

$b^{2} \sqrt[3]{20 b} \sqrt[3]{27 (6) b^{3}}$

Step 3.2

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

$b^{2} \sqrt[3]{20 b} \sqrt[3]{3^{3} \cdot 6 b^{3}}$

Step 3.3

Move $6$ .

$b^{2} \sqrt[3]{20 b} \sqrt[3]{3^{3} b^{3} \cdot 6}$

Step 3.4

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

$b^{2} \sqrt[3]{20 b} \sqrt[3]{{(3 b)}^{3} \cdot 6}$

Step 4

Pull terms out from under the radical.

$b^{2} \sqrt[3]{20 b} (3 b \sqrt[3]{6})$

Step 5

Rewrite using the commutative property of multiplication.

$3 \sqrt[3]{6} b^{2} \sqrt[3]{20 b} b$

Step 6

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

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Step 6.1

Move $b$ .

$3 \sqrt[3]{6} (b \cdot b^{2}) \sqrt[3]{20 b}$

Step 6.2

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

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Step 6.2.1

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

$3 \sqrt[3]{6} (b^{1} b^{2}) \sqrt[3]{20 b}$

Step 6.2.2

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

$3 \sqrt[3]{6} b^{1 + 2} \sqrt[3]{20 b}$

Step 6.3

Add $1$ and $2$ .

$3 \sqrt[3]{6} b^{3} \sqrt[3]{20 b}$

Step 7

Multiply $3 \sqrt[3]{6} b^{3} \sqrt[3]{20 b}$ .

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Step 7.1

Combine using the product rule for radicals.

$3 b^{3} \sqrt[3]{6 (20 b)}$

Step 7.2

Multiply $6$ by $20$ .

$3 b^{3} \sqrt[3]{120 b}$

Step 8

Rewrite $120 b$ as $2^{3} \cdot (15 b)$ .

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Step 8.1

Factor $8$ out of $120$ .

$3 b^{3} \sqrt[3]{8 (15) b}$

Step 8.2

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

$3 b^{3} \sqrt[3]{2^{3} \cdot 15 b}$

Step 8.3

Add parentheses.

$3 b^{3} \sqrt[3]{2^{3} \cdot (15 b)}$

Step 9

Pull terms out from under the radical.

$3 b^{3} (2 \sqrt[3]{15 b})$

Step 10

Multiply $2$ by $3$ .

$6 b^{3} \sqrt[3]{15 b}$