Algebra Examples

$(\sqrt[3]{405 a^{4}}) (\sqrt[3]{54 a})$

Step 1

Rewrite $405 a^{4}$ as ${(3 a)}^{3} \cdot (15 a)$ .

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Step 1.1

Factor $27$ out of $405$ .

$\sqrt[3]{27 (15) a^{4}} \sqrt[3]{54 a}$

Step 1.2

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

$\sqrt[3]{3^{3} \cdot 15 a^{4}} \sqrt[3]{54 a}$

Step 1.3

Factor out $a^{3}$ .

$\sqrt[3]{3^{3} \cdot 15 (a^{3} a)} \sqrt[3]{54 a}$

Step 1.4

Move $15$ .

$\sqrt[3]{3^{3} a^{3} \cdot 15 a} \sqrt[3]{54 a}$

Step 1.5

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

$\sqrt[3]{{(3 a)}^{3} \cdot 15 a} \sqrt[3]{54 a}$

Step 1.6

Add parentheses.

$\sqrt[3]{{(3 a)}^{3} \cdot (15 a)} \sqrt[3]{54 a}$

Step 2

Pull terms out from under the radical.

$3 a \sqrt[3]{15 a} \sqrt[3]{54 a}$

Step 3

Rewrite $54 a$ as $3^{3} \cdot (2 a)$ .

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Step 3.1

Factor $27$ out of $54$ .

$3 a \sqrt[3]{15 a} \sqrt[3]{27 (2) a}$

Step 3.2

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

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

Step 3.3

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $3 a \sqrt[3]{15 a} (3 \sqrt[3]{2 a})$ .

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Step 5.1

Multiply $3$ by $3$ .

$9 a \sqrt[3]{15 a} \sqrt[3]{2 a}$

Step 5.2

Combine using the product rule for radicals.

$9 a \sqrt[3]{2 a (15 a)}$

Step 5.3

Multiply $15$ by $2$ .

$9 a \sqrt[3]{30 a \cdot a}$

Step 5.4

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

$9 a \sqrt[3]{30 (a^{1} a)}$

Step 5.5

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

$9 a \sqrt[3]{30 (a^{1} a^{1})}$

Step 5.6

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

$9 a \sqrt[3]{30 a^{1 + 1}}$

Step 5.7

Add $1$ and $1$ .

$9 a \sqrt[3]{30 a^{2}}$