Precalculus Examples

$\frac{\sqrt[4]{27}}{\sqrt[3]{81}}$

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $27$ out of $81$ .

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

Step 1.1.2

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

$\frac{\sqrt[4]{27}}{\sqrt[3]{3^{3} \cdot 3}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt[4]{27}}{3 \sqrt[3]{3}}$

Step 2

Multiply $\frac{\sqrt[4]{27}}{3 \sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$\frac{\sqrt[4]{27}}{3 \sqrt[3]{3}} \cdot \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[4]{27}}{3 \sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 \sqrt[3]{3} {\sqrt[3]{3}}^{2}}$

Step 3.2

Move $\sqrt[3]{3}$ .

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 (\sqrt[3]{3} {\sqrt[3]{3}}^{2})}$

Step 3.3

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

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 ({\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2})}$

Step 3.4

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

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 {\sqrt[3]{3}}^{1 + 2}}$

Step 3.5

Add $1$ and $2$ .

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 {\sqrt[3]{3}}^{3}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 {(3^{\frac{1}{3}})}^{3}}$

Step 3.6.2

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

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 \cdot 3^{\frac{1}{3} \cdot 3}}$

Step 3.6.3

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

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 \cdot 3^{\frac{3}{3}}}$

Step 3.6.4

Cancel the common factor of $3$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 \cdot 3^{\frac{3}{3}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 \cdot 3^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{\sqrt[4]{27} {\sqrt[3]{3}}^{2}}{3 \cdot 3}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{\sqrt[4]{27} \sqrt[3]{3^{2}}}{3 \cdot 3}$

Step 4.2

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

$\frac{\sqrt[4]{27} \sqrt[3]{9}}{3 \cdot 3}$

Step 5

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{27} \sqrt[3]{9}}{9}$

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{27^{\frac{1}{4}} \sqrt[3]{9}}{9}$

Step 6.1.2

Rewrite $27^{\frac{1}{4}}$ as $27^{\frac{3}{12}}$ .

$\frac{27^{\frac{3}{12}} \sqrt[3]{9}}{9}$

Step 6.1.3

Rewrite $27^{\frac{3}{12}}$ as $\sqrt[12]{27^{3}}$ .

$\frac{\sqrt[12]{27^{3}} \sqrt[3]{9}}{9}$

Step 6.1.4

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

$\frac{\sqrt[12]{27^{3}} \cdot 9^{\frac{1}{3}}}{9}$

Step 6.1.5

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

$\frac{\sqrt[12]{27^{3}} \cdot 9^{\frac{4}{12}}}{9}$

Step 6.1.6

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

$\frac{\sqrt[12]{27^{3}} \sqrt[12]{9^{4}}}{9}$

Step 6.2

Combine using the product rule for radicals.

$\frac{\sqrt[12]{27^{3} \cdot 9^{4}}}{9}$

Step 6.3

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

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Multiply $3$ by $3$ .

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

Step 6.5

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

$\frac{\sqrt[12]{3^{9} \cdot {(3^{2})}^{4}}}{9}$

Step 6.6

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

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Step 6.6.1

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

$\frac{\sqrt[12]{3^{9} \cdot 3^{2 \cdot 4}}}{9}$

Step 6.6.2

Multiply $2$ by $4$ .

$\frac{\sqrt[12]{3^{9} \cdot 3^{8}}}{9}$

Step 6.7

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

$\frac{\sqrt[12]{3^{9 + 8}}}{9}$

Step 6.8

Add $9$ and $8$ .

$\frac{\sqrt[12]{3^{17}}}{9}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $3$ to the power of $17$ .

$\frac{\sqrt[12]{129140163}}{9}$

Step 7.2

Rewrite $129140163$ as $3^{12} \cdot 243$ .

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Step 7.2.1

Factor $531441$ out of $129140163$ .

$\frac{\sqrt[12]{531441 (243)}}{9}$

Step 7.2.2

Rewrite $531441$ as $3^{12}$ .

$\frac{\sqrt[12]{3^{12} \cdot 243}}{9}$

Step 7.3

Pull terms out from under the radical.

$\frac{3 \sqrt[12]{243}}{9}$

Step 8

Cancel the common factor of $3$ and $9$ .

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Step 8.1

Factor $3$ out of $3 \sqrt[12]{243}$ .

$\frac{3 (\sqrt[12]{243})}{9}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $3$ out of $9$ .

$\frac{3 \sqrt[12]{243}}{3 \cdot 3}$

Step 8.2.2

Cancel the common factor.

$\frac{3 \sqrt[12]{243}}{3 \cdot 3}$

Step 8.2.3

Rewrite the expression.

$\frac{\sqrt[12]{243}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[12]{243}}{3}$

Decimal Form:

$0.52684063 \dots$