Algebra Examples

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

Step 1

Simplify the numerator.

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

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

Step 1.1.6

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

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

Step 1.2

Combine using the product rule for radicals.

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

Step 1.3

Raise $3$ to the power of $9$ .

$\frac{\sqrt[12]{19683 x^{8}}}{\sqrt{3 x}}$

Step 2

Multiply $\frac{\sqrt[12]{19683 x^{8}}}{\sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\frac{\sqrt[12]{19683 x^{8}}}{\sqrt{3 x}} \cdot \frac{\sqrt{3 x}}{\sqrt{3 x}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[12]{19683 x^{8}}}{\sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{\sqrt{3 x} \sqrt{3 x}}$

Step 3.2

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

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{\sqrt{3 x}}^{1} \sqrt{3 x}}$

Step 3.3

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

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1}}$

Step 3.4

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

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{\sqrt{3 x}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{\sqrt{3 x}}^{2}}$

Step 3.6

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

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Step 3.6.1

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

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{({(3 x)}^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{(3 x)}^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{(3 x)}^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{(3 x)}^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{{(3 x)}^{1}}$

Step 3.6.5

Simplify.

$\frac{\sqrt[12]{19683 x^{8}} \sqrt{3 x}}{3 x}$

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 $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\frac{\sqrt[12]{19683 x^{8}} {(3 x)}^{\frac{1}{2}}}{3 x}$

Step 4.1.2

Rewrite ${(3 x)}^{\frac{1}{2}}$ as ${(3 x)}^{\frac{6}{12}}$ .

$\frac{\sqrt[12]{19683 x^{8}} {(3 x)}^{\frac{6}{12}}}{3 x}$

Step 4.1.3

Rewrite ${(3 x)}^{\frac{6}{12}}$ as $\sqrt[12]{{(3 x)}^{6}}$ .

$\frac{\sqrt[12]{19683 x^{8}} \sqrt[12]{{(3 x)}^{6}}}{3 x}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[12]{19683 x^{8} {(3 x)}^{6}}}{3 x}$

Step 4.3

Apply the product rule to $3 x$ .

$\frac{\sqrt[12]{19683 x^{8} \cdot 3^{6} x^{6}}}{3 x}$

Step 4.4

Combine exponents.

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Step 4.4.1

Rewrite $19683$ as $3^{9}$ .

$\frac{\sqrt[12]{3^{6} \cdot 3^{9} x^{8} x^{6}}}{3 x}$

Step 4.4.2

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

$\frac{\sqrt[12]{3^{6 + 9} x^{8} x^{6}}}{3 x}$

Step 4.4.3

Add $6$ and $9$ .

$\frac{\sqrt[12]{3^{15} x^{8} x^{6}}}{3 x}$

Step 4.4.4

Multiply $x^{8}$ by $x^{6}$ by adding the exponents.

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Step 4.4.4.1

Move $x^{6}$ .

$\frac{\sqrt[12]{3^{15} (x^{6} x^{8})}}{3 x}$

Step 4.4.4.2

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

$\frac{\sqrt[12]{3^{15} x^{6 + 8}}}{3 x}$

Step 4.4.4.3

Add $6$ and $8$ .

$\frac{\sqrt[12]{3^{15} x^{14}}}{3 x}$

Step 4.5

Raise $3$ to the power of $15$ .

$\frac{\sqrt[12]{14348907 x^{14}}}{3 x}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $14348907 x^{14}$ as ${(3 x)}^{12} \cdot (27 x^{2})$ .

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Step 5.1.1

Factor $531441$ out of $14348907$ .

$\frac{\sqrt[12]{531441 (27) x^{14}}}{3 x}$

Step 5.1.2

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

$\frac{\sqrt[12]{3^{12} \cdot 27 x^{14}}}{3 x}$

Step 5.1.3

Factor out $x^{12}$ .

$\frac{\sqrt[12]{3^{12} \cdot 27 (x^{12} x^{2})}}{3 x}$

Step 5.1.4

Move $27$ .

$\frac{\sqrt[12]{3^{12} x^{12} \cdot 27 x^{2}}}{3 x}$

Step 5.1.5

Rewrite $3^{12} x^{12}$ as ${(3 x)}^{12}$ .

$\frac{\sqrt[12]{{(3 x)}^{12} \cdot 27 x^{2}}}{3 x}$

Step 5.1.6

Add parentheses.

$\frac{\sqrt[12]{{(3 x)}^{12} \cdot (27 x^{2})}}{3 x}$

Step 5.2

Pull terms out from under the radical.

$\frac{3 x \sqrt[12]{27 x^{2}}}{3 x}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $3$ .

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Step 6.1.1

Cancel the common factor.

$\frac{3 x \sqrt[12]{27 x^{2}}}{3 x}$

Step 6.1.2

Rewrite the expression.

$\frac{x \sqrt[12]{27 x^{2}}}{x}$

Step 6.2

Cancel the common factor of $x$ .

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Step 6.2.1

Cancel the common factor.

$\frac{x \sqrt[12]{27 x^{2}}}{x}$

Step 6.2.2

Divide $\sqrt[12]{27 x^{2}}$ by $1$ .

$\sqrt[12]{27 x^{2}}$