Algebra Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.2

Combine using the product rule for radicals.

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

Step 1.3

Multiply $3^{4}$ by $3^{3}$ by adding the exponents.

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Step 1.3.1

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

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

Step 1.3.2

Add $4$ and $3$ .

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

Step 2

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

${\sqrt[12]{2187}}^{12}$

Step 3

Rewrite ${\sqrt[12]{2187}}^{12}$ as $2187$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[12]{2187}$ as $2187^{\frac{1}{12}}$ .

${(2187^{\frac{1}{12}})}^{12}$

Step 3.2

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

$2187^{\frac{1}{12} \cdot 12}$

Step 3.3

Combine $\frac{1}{12}$ and $12$ .

$2187^{\frac{12}{12}}$

Step 3.4

Cancel the common factor of $12$ .

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Step 3.4.1

Cancel the common factor.

$2187^{\frac{12}{12}}$

Step 3.4.2

Rewrite the expression.

$2187^{1}$

Step 3.5

Evaluate the exponent.

$2187$