Algebra Examples

$\sqrt{27} \cdot \sqrt[3]{9}$

Step 1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1

Factor $9$ out of $27$ .

$\sqrt{9 (3)} \cdot \sqrt[3]{9}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 3} \cdot \sqrt[3]{9}$

Step 2

Pull terms out from under the radical.

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

Step 3

Multiply $3 \sqrt{3} \sqrt[3]{9}$ .

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Step 3.1

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

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$3 (9^{\frac{1}{3}} \sqrt{3})$

Step 3.1.2

Rewrite $9^{\frac{1}{3}}$ as $9^{\frac{2}{6}}$ .

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

Step 3.1.3

Rewrite $9^{\frac{2}{6}}$ as $\sqrt[6]{9^{2}}$ .

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

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

Combine using the product rule for radicals.

$3 \sqrt[6]{9^{2} \cdot 3^{3}}$

Step 3.3

Rewrite $9$ as $3^{2}$ .

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

Step 3.4

Multiply the exponents in ${(3^{2})}^{2}$ .

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Step 3.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.4.2

Multiply $2$ by $2$ .

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

Step 3.5

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

$3 \sqrt[6]{3^{4 + 3}}$

Step 3.6

Add $4$ and $3$ .

$3 \sqrt[6]{3^{7}}$

Step 4

Raise $3$ to the power of $7$ .

$3 \sqrt[6]{2187}$

Step 5

Rewrite $2187$ as $3^{6} \cdot 3$ .

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Step 5.1

Factor $729$ out of $2187$ .

$3 \sqrt[6]{729 (3)}$

Step 5.2

Rewrite $729$ as $3^{6}$ .

$3 \sqrt[6]{3^{6} \cdot 3}$

Step 6

Pull terms out from under the radical.

$3 (3 \sqrt[6]{3})$

Step 7

Multiply $3$ by $3$ .

$9 \sqrt[6]{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$9 \sqrt[6]{3}$

Decimal Form:

$10.80843259 \dots$