Algebra Examples

$\sqrt[3]{3} \cdot \sqrt[5]{3^{6}}$

Step 1

Raise $3$ to the power of $6$ .

$\sqrt[3]{3} \cdot \sqrt[5]{729}$

Step 2

Rewrite $729$ as $3^{5} \cdot 3$ .

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Step 2.1

Factor $243$ out of $729$ .

$\sqrt[3]{3} \cdot \sqrt[5]{243 (3)}$

Step 2.2

Rewrite $243$ as $3^{5}$ .

$\sqrt[3]{3} \cdot \sqrt[5]{3^{5} \cdot 3}$

Step 3

Pull terms out from under the radical.

$\sqrt[3]{3} \cdot (3 \sqrt[5]{3})$

Step 4

Multiply $\sqrt[3]{3} (3 \sqrt[5]{3})$ .

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Step 4.1

Rewrite the expression using the least common index of $15$ .

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Step 4.1.1

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

$3 (3^{\frac{1}{3}} \sqrt[5]{3})$

Step 4.1.2

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{5}{15}}$ .

$3 (3^{\frac{5}{15}} \sqrt[5]{3})$

Step 4.1.3

Rewrite $3^{\frac{5}{15}}$ as $\sqrt[15]{3^{5}}$ .

$3 (\sqrt[15]{3^{5}} \sqrt[5]{3})$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3}$ as $3^{\frac{1}{5}}$ .

$3 (\sqrt[15]{3^{5}} \cdot 3^{\frac{1}{5}})$

Step 4.1.5

Rewrite $3^{\frac{1}{5}}$ as $3^{\frac{3}{15}}$ .

$3 (\sqrt[15]{3^{5}} \cdot 3^{\frac{3}{15}})$

Step 4.1.6

Rewrite $3^{\frac{3}{15}}$ as $\sqrt[15]{3^{3}}$ .

$3 (\sqrt[15]{3^{5}} \sqrt[15]{3^{3}})$

Step 4.2

Combine using the product rule for radicals.

$3 \sqrt[15]{3^{5} \cdot 3^{3}}$

Step 4.3

Multiply $3^{5}$ by $3^{3}$ by adding the exponents.

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Step 4.3.1

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

$3 \sqrt[15]{3^{5 + 3}}$

Step 4.3.2

Add $5$ and $3$ .

$3 \sqrt[15]{3^{8}}$

Step 5

Raise $3$ to the power of $8$ .

$3 \sqrt[15]{6561}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt[15]{6561}$

Decimal Form:

$5.38996473 \dots$