Algebra Examples

$\sqrt[3]{6} \div \sqrt[4]{6}$

Step 1

Rewrite the division as a fraction.

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

Step 2

Multiply $\frac{\sqrt[3]{6}}{\sqrt[4]{6}}$ by $\frac{{\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{3}}$ .

$\frac{\sqrt[3]{6}}{\sqrt[4]{6}} \cdot \frac{{\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{3}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[3]{6}}{\sqrt[4]{6}}$ by $\frac{{\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{3}}$ .

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

Step 3.2

Raise $\sqrt[4]{6}$ to the power of $1$ .

$\frac{\sqrt[3]{6} {\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{1} {\sqrt[4]{6}}^{3}}$

Step 3.3

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

$\frac{\sqrt[3]{6} {\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{1 + 3}}$

Step 3.4

Add $1$ and $3$ .

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

Step 3.5

Rewrite ${\sqrt[4]{6}}^{4}$ as $6$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{6}$ as $6^{\frac{1}{4}}$ .

$\frac{\sqrt[3]{6} {\sqrt[4]{6}}^{3}}{{(6^{\frac{1}{4}})}^{4}}$

Step 3.5.2

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

$\frac{\sqrt[3]{6} {\sqrt[4]{6}}^{3}}{6^{\frac{1}{4} \cdot 4}}$

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{6} {\sqrt[4]{6}}^{3}}{6^{1}}$

Step 3.5.5

Evaluate the exponent.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{6}}^{3}$ as $\sqrt[4]{6^{3}}$ .

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

Step 4.2

Raise $6$ to the power of $3$ .

$\frac{\sqrt[3]{6} \sqrt[4]{216}}{6}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$\frac{6^{\frac{1}{3}} \sqrt[4]{216}}{6}$

Step 5.1.2

Rewrite $6^{\frac{1}{3}}$ as $6^{\frac{4}{12}}$ .

$\frac{6^{\frac{4}{12}} \sqrt[4]{216}}{6}$

Step 5.1.3

Rewrite $6^{\frac{4}{12}}$ as $\sqrt[12]{6^{4}}$ .

$\frac{\sqrt[12]{6^{4}} \sqrt[4]{216}}{6}$

Step 5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{216}$ as $216^{\frac{1}{4}}$ .

$\frac{\sqrt[12]{6^{4}} \cdot 216^{\frac{1}{4}}}{6}$

Step 5.1.5

Rewrite $216^{\frac{1}{4}}$ as $216^{\frac{3}{12}}$ .

$\frac{\sqrt[12]{6^{4}} \cdot 216^{\frac{3}{12}}}{6}$

Step 5.1.6

Rewrite $216^{\frac{3}{12}}$ as $\sqrt[12]{216^{3}}$ .

$\frac{\sqrt[12]{6^{4}} \sqrt[12]{216^{3}}}{6}$

Step 5.2

Combine using the product rule for radicals.

$\frac{\sqrt[12]{6^{4} \cdot 216^{3}}}{6}$

Step 5.3

Rewrite $216$ as $6^{3}$ .

$\frac{\sqrt[12]{6^{4} \cdot {(6^{3})}^{3}}}{6}$

Step 5.4

Multiply the exponents in ${(6^{3})}^{3}$ .

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Step 5.4.1

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

$\frac{\sqrt[12]{6^{4} \cdot 6^{3 \cdot 3}}}{6}$

Step 5.4.2

Multiply $3$ by $3$ .

$\frac{\sqrt[12]{6^{4} \cdot 6^{9}}}{6}$

Step 5.5

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

$\frac{\sqrt[12]{6^{4 + 9}}}{6}$

Step 5.6

Add $4$ and $9$ .

$\frac{\sqrt[12]{6^{13}}}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Raise $6$ to the power of $13$ .

$\frac{\sqrt[12]{13060694016}}{6}$

Step 6.2

Rewrite $13060694016$ as $6^{12} \cdot 6$ .

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Step 6.2.1

Factor $2176782336$ out of $13060694016$ .

$\frac{\sqrt[12]{2176782336 (6)}}{6}$

Step 6.2.2

Rewrite $2176782336$ as $6^{12}$ .

$\frac{\sqrt[12]{6^{12} \cdot 6}}{6}$

Step 6.3

Pull terms out from under the radical.

$\frac{6 \sqrt[12]{6}}{6}$

Step 7

Cancel the common factor of $6$ .

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Step 7.1

Cancel the common factor.

$\frac{6 \sqrt[12]{6}}{6}$

Step 7.2

Divide $\sqrt[12]{6}$ by $1$ .

$\sqrt[12]{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt[12]{6}$

Decimal Form:

$1.16103667 \dots$