Algebra Examples

$\sqrt[4]{3^{5}} \cdot \sqrt[3]{3^{4}}$

Step 1

Raise $3$ to the power of $5$ .

$\sqrt[4]{243} \cdot \sqrt[3]{3^{4}}$

Step 2

Rewrite $243$ as $3^{4} \cdot 3$ .

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Step 2.1

Factor $81$ out of $243$ .

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

Step 2.2

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

Raise $3$ to the power of $4$ .

$3 \sqrt[4]{3} \cdot \sqrt[3]{81}$

Step 5

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 5.1

Factor $27$ out of $81$ .

$3 \sqrt[4]{3} \cdot \sqrt[3]{27 (3)}$

Step 5.2

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

Multiply $3 \sqrt[4]{3} (3 \sqrt[3]{3})$ .

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Step 7.1

Multiply $3$ by $3$ .

$9 \sqrt[4]{3} \sqrt[3]{3}$

Step 7.2

Rewrite the expression using the least common index of $12$ .

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Step 7.2.1

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

$9 (3^{\frac{1}{3}} \sqrt[4]{3})$

Step 7.2.2

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{4}{12}}$ .

$9 (3^{\frac{4}{12}} \sqrt[4]{3})$

Step 7.2.3

Rewrite $3^{\frac{4}{12}}$ as $\sqrt[12]{3^{4}}$ .

$9 (\sqrt[12]{3^{4}} \sqrt[4]{3})$

Step 7.2.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$9 (\sqrt[12]{3^{4}} \cdot 3^{\frac{1}{4}})$

Step 7.2.5

Rewrite $3^{\frac{1}{4}}$ as $3^{\frac{3}{12}}$ .

$9 (\sqrt[12]{3^{4}} \cdot 3^{\frac{3}{12}})$

Step 7.2.6

Rewrite $3^{\frac{3}{12}}$ as $\sqrt[12]{3^{3}}$ .

$9 (\sqrt[12]{3^{4}} \sqrt[12]{3^{3}})$

Step 7.3

Combine using the product rule for radicals.

$9 \sqrt[12]{3^{4} \cdot 3^{3}}$

Step 7.4

Multiply $3^{4}$ by $3^{3}$ 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.

$9 \sqrt[12]{3^{4 + 3}}$

Step 7.4.2

Add $4$ and $3$ .

$9 \sqrt[12]{3^{7}}$

Step 8

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

$9 \sqrt[12]{2187}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$9 \sqrt[12]{2187}$

Decimal Form:

$17.08296461 \dots$