Algebra Examples

$\sqrt{12 x} \div \sqrt[4]{3 x}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{12 x}}{\sqrt[4]{3 x}}$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $12$ .

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

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.1.3

Add parentheses.

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

Raise $\sqrt[4]{3 x}$ to the power of $1$ .

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

Step 4.3

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

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

Step 4.4

Add $1$ and $3$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.5.4

Cancel the common factor of $4$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[4]{3 x}}^{3}$ as $\sqrt[4]{{(3 x)}^{3}}$ .

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

Step 5.2

Apply the product rule to $3 x$ .

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

Step 5.3

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

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

Step 5.4

Combine exponents.

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Step 5.4.1

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

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Step 5.4.1.1

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

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

Step 5.4.1.2

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

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

Step 5.4.1.3

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

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

Step 5.4.2

Combine using the product rule for radicals.

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

Step 5.4.3

Apply the product rule to $3 x$ .

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

Step 5.4.4

Combine exponents.

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Step 5.4.4.1

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

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

Step 5.4.4.2

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

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

Step 5.4.4.3

Add $2$ and $3$ .

$\frac{2 \sqrt[4]{3^{5} x^{3} x^{2}}}{3 x}$

Step 5.4.4.4

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

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Step 5.4.4.4.1

Move $x^{2}$ .

$\frac{2 \sqrt[4]{3^{5} (x^{2} x^{3})}}{3 x}$

Step 5.4.4.4.2

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

$\frac{2 \sqrt[4]{3^{5} x^{2 + 3}}}{3 x}$

Step 5.4.4.4.3

Add $2$ and $3$ .

$\frac{2 \sqrt[4]{3^{5} x^{5}}}{3 x}$

Step 5.4.5

Raise $3$ to the power of $5$ .

$\frac{2 \sqrt[4]{243 x^{5}}}{3 x}$

Step 5.5

Rewrite $243 x^{5}$ as ${(3 x)}^{4} \cdot (3 x)$ .

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Step 5.5.1

Factor $81$ out of $243$ .

$\frac{2 \sqrt[4]{81 (3) x^{5}}}{3 x}$

Step 5.5.2

Rewrite $81$ as $3^{4}$ .

$\frac{2 \sqrt[4]{3^{4} \cdot 3 x^{5}}}{3 x}$

Step 5.5.3

Factor out $x^{4}$ .

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

Step 5.5.4

Move $3$ .

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

Step 5.5.5

Rewrite $3^{4} x^{4}$ as ${(3 x)}^{4}$ .

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

Step 5.5.6

Add parentheses.

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

Step 5.6

Pull terms out from under the radical.

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

Step 6

Multiply $2$ by $3$ .

$\frac{6 x \sqrt[4]{3 x}}{3 x}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $6$ and $3$ .

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Step 7.1.1

Factor $3$ out of $6 x \sqrt[4]{3 x}$ .

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

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $3$ out of $3 x$ .

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

Step 7.1.2.2

Cancel the common factor.

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

Step 7.1.2.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Divide $2 \sqrt[4]{3 x}$ by $1$ .

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