Algebra Examples

$\sqrt[4]{81 x^{2}} \cdot \sqrt[3]{27 x^{7}}$

Step 1

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

$\sqrt[4]{3^{4} x^{2}} \cdot \sqrt[3]{27 x^{7}}$

Step 2

Pull terms out from under the radical.

$3 \sqrt[4]{x^{2}} \cdot \sqrt[3]{27 x^{7}}$

Step 3

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$3 \sqrt{\sqrt{x^{2}}} \cdot \sqrt[3]{27 x^{7}}$

Step 4

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

$3 \sqrt{x} \cdot \sqrt[3]{27 x^{7}}$

Step 5

Rewrite $27 x^{7}$ as ${(3 x^{2})}^{3} x$ .

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Step 5.1

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

$3 \sqrt{x} \cdot \sqrt[3]{3^{3} x^{7}}$

Step 5.2

Factor out $x^{6}$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

Multiply $3 \sqrt{x} (3 x^{2} \sqrt[3]{x})$ .

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Step 7.1

Multiply $3$ by $3$ .

$9 \sqrt{x} (x^{2} \sqrt[3]{x})$

Step 7.2

Rewrite the expression using the least common index of $6$ .

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Step 7.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$9 (x^{2} (x^{\frac{1}{2}} \sqrt[3]{x}))$

Step 7.2.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$9 (x^{2} (x^{\frac{3}{6}} \sqrt[3]{x}))$

Step 7.2.3

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$9 (x^{2} (\sqrt[6]{x^{3}} \sqrt[3]{x}))$

Step 7.2.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$9 (x^{2} (\sqrt[6]{x^{3}} x^{\frac{1}{3}}))$

Step 7.2.5

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

$9 (x^{2} (\sqrt[6]{x^{3}} x^{\frac{2}{6}}))$

Step 7.2.6

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

$9 (x^{2} (\sqrt[6]{x^{3}} \sqrt[6]{x^{2}}))$

Step 7.3

Combine using the product rule for radicals.

$9 (x^{2} \sqrt[6]{x^{3} x^{2}})$

Step 7.4

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 7.4.1

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

$9 (x^{2} \sqrt[6]{x^{3 + 2}})$

Step 7.4.2

Add $3$ and $2$ .

$9 (x^{2} \sqrt[6]{x^{5}})$

$9 x^{2} \sqrt[6]{x^{5}}$