Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1.1

Factor $9$ out of $27$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

Raise $\sqrt[4]{27}$ to the power of $1$ .

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

Step 3.1.3

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

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

Step 3.1.4

Add $1$ and $3$ .

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

Step 3.1.5

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 3.1.5.4

Cancel the common factor of $4$ .

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Step 3.1.5.4.1

Cancel the common factor.

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

Step 3.1.5.4.2

Rewrite the expression.

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

Step 3.1.5.5

Evaluate the exponent.

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

Step 3.2

Cancel the common factor of $3$ and $27$ .

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Step 3.2.1

Factor $3$ out of $3 \sqrt{3} {\sqrt[4]{27}}^{3}$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $3$ out of $27$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{27}}^{3}$ as $\sqrt[4]{27^{3}}$ .

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

Step 4.2

Raise $27$ to the power of $3$ .

$\frac{\sqrt{3} \sqrt[4]{19683}}{9}$

Step 4.3

Rewrite $19683$ as $9^{4} \cdot 3$ .

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Step 4.3.1

Factor $6561$ out of $19683$ .

$\frac{\sqrt{3} \sqrt[4]{6561 (3)}}{9}$

Step 4.3.2

Rewrite $6561$ as $9^{4}$ .

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

Step 4.4

Pull terms out from under the radical.

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

Step 4.5

Combine exponents.

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Step 4.5.1

Rewrite the expression using the least common index of $4$ .

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Step 4.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{9 (3^{\frac{1}{2}} \sqrt[4]{3})}{9}$

Step 4.5.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

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

Step 4.5.1.3

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

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

Step 4.5.2

Combine using the product rule for radicals.

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

Step 4.5.3

Multiply $3^{2}$ by $3$ by adding the exponents.

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Step 4.5.3.1

Multiply $3^{2}$ by $3$ .

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Step 4.5.3.1.1

Raise $3$ to the power of $1$ .

$\frac{9 \sqrt[4]{3^{2} \cdot 3^{1}}}{9}$

Step 4.5.3.1.2

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

$\frac{9 \sqrt[4]{3^{2 + 1}}}{9}$

Step 4.5.3.2

Add $2$ and $1$ .

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

Step 4.6

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

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

Step 5

Cancel the common factor of $9$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $\sqrt[4]{27}$ by $1$ .

$\sqrt[4]{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[4]{27}$

Decimal Form:

$2.27950705 \dots$