Algebra Examples

$\sqrt[3]{27 x^{12}} \cdot \sqrt[4]{81 x^{11}}$

Step 1

Rewrite $27 x^{12}$ as ${(3 x^{4})}^{3}$ .

$\sqrt[3]{{(3 x^{4})}^{3}} \cdot \sqrt[4]{81 x^{11}}$

Step 2

Pull terms out from under the radical, assuming real numbers.

$3 x^{4} \cdot \sqrt[4]{81 x^{11}}$

Step 3

Rewrite $81 x^{11}$ as ${(3 x^{2})}^{4} x^{3}$ .

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Step 3.1

Rewrite $81$ as $3^{4}$ .

$3 x^{4} \cdot \sqrt[4]{3^{4} x^{11}}$

Step 3.2

Factor out $x^{8}$ .

$3 x^{4} \cdot \sqrt[4]{3^{4} (x^{8} x^{3})}$

Step 3.3

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

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

Step 3.4

Rewrite $3^{4} {(x^{2})}^{4}$ as ${(3 x^{2})}^{4}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 5.1

Move $x^{2}$ .

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

Step 5.2

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

$3 x^{2 + 4} \cdot (3 \sqrt[4]{x^{3}})$

Step 5.3

Add $2$ and $4$ .

$3 x^{6} \cdot (3 \sqrt[4]{x^{3}})$

Step 6

Multiply $3$ by $3$ .

$9 x^{6} \sqrt[4]{x^{3}}$