Algebra Examples

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

Step 1

Multiply $\frac{\sqrt[4]{3}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$\frac{\sqrt[4]{3}}{\sqrt[3]{3}} \cdot \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[4]{3}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}}$

Step 2.2

Raise $\sqrt[3]{3}$ to the power of $1$ .

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}$

Step 2.3

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

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{3}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{{(3^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{3^{1}}$

Step 2.5.5

Evaluate the exponent.

$\frac{\sqrt[4]{3} {\sqrt[3]{3}}^{2}}{3}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{3}}^{2}$ as $\sqrt[3]{3^{2}}$ .

$\frac{\sqrt[4]{3} \sqrt[3]{3^{2}}}{3}$

Step 3.2

Raise $3$ to the power of $2$ .

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Rewrite $9^{\frac{1}{3}}$ as $9^{\frac{4}{12}}$ .

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

Step 4.1.6

Rewrite $9^{\frac{4}{12}}$ as $\sqrt[12]{9^{4}}$ .

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

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

$\frac{\sqrt[12]{3^{3} \cdot {(3^{2})}^{4}}}{3}$

Step 4.4

Multiply the exponents in ${(3^{2})}^{4}$ .

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Step 4.4.1

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

$\frac{\sqrt[12]{3^{3} \cdot 3^{2 \cdot 4}}}{3}$

Step 4.4.2

Multiply $2$ by $4$ .

$\frac{\sqrt[12]{3^{3} \cdot 3^{8}}}{3}$

Step 4.5

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

$\frac{\sqrt[12]{3^{3 + 8}}}{3}$

Step 4.6

Add $3$ and $8$ .

$\frac{\sqrt[12]{3^{11}}}{3}$

Step 5

Raise $3$ to the power of $11$ .

$\frac{\sqrt[12]{177147}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[12]{177147}}{3}$

Decimal Form:

$0.91251475 \dots$