Basic Math Examples

$\frac{\sqrt{n}}{\sqrt[3]{n}}$

Step 1

Multiply

$\frac{\sqrt{n}}{\sqrt[3]{n}}$ by

$\frac{{\sqrt[3]{n}}^{2}}{{\sqrt[3]{n}}^{2}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply

$\frac{\sqrt{n}}{\sqrt[3]{n}}$ by

$\frac{{\sqrt[3]{n}}^{2}}{{\sqrt[3]{n}}^{2}}$ .

$\frac{\sqrt{n} {\sqrt[3]{n}}^{2}}{\sqrt[3]{n} {\sqrt[3]{n}}^{2}}$

Step 2.2

Raise

$\sqrt[3]{n}$ to the power of

$1$ .

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

Step 2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.4

Add

$1$ and

$2$ .

$\frac{\sqrt{n} {\sqrt[3]{n}}^{2}}{{\sqrt[3]{n}}^{3}}$

Step 2.5

Rewrite

${\sqrt[3]{n}}^{3}$ as

$n$ .

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Step 2.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{n}$ as

$n^{\frac{1}{3}}$ .

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

Step 2.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5.3

Combine

$\frac{1}{3}$ and

$3$ .

$\frac{\sqrt{n} {\sqrt[3]{n}}^{2}}{n^{\frac{3}{3}}}$

Step 2.5.4

Cancel the common factor of

$3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt{n} {\sqrt[3]{n}}^{2}}{n^{\frac{3}{3}}}$

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

$\frac{\sqrt{n} {\sqrt[3]{n}}^{2}}{n}$

Step 3

Rewrite

${\sqrt[3]{n}}^{2}$ as

$\sqrt[3]{n^{2}}$ .

$\frac{\sqrt{n} \sqrt[3]{n^{2}}}{n}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of

$6$ .

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Step 4.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{n}$ as

$n^{\frac{1}{2}}$ .

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

Step 4.1.2

Rewrite

$n^{\frac{1}{2}}$ as

$n^{\frac{3}{6}}$ .

$\frac{n^{\frac{3}{6}} \sqrt[3]{n^{2}}}{n}$

Step 4.1.3

Rewrite

$n^{\frac{3}{6}}$ as

$\sqrt[6]{n^{3}}$ .

$\frac{\sqrt[6]{n^{3}} \sqrt[3]{n^{2}}}{n}$

Step 4.1.4

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{n^{2}}$ as

$n^{\frac{2}{3}}$ .

$\frac{\sqrt[6]{n^{3}} n^{\frac{2}{3}}}{n}$

Step 4.1.5

Rewrite

$n^{\frac{2}{3}}$ as

$n^{\frac{4}{6}}$ .

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

Step 4.1.6

Rewrite

$n^{\frac{4}{6}}$ as

$\sqrt[6]{n^{4}}$ .

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Multiply

$n^{3}$ by

$n^{4}$ by adding the exponents.

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Step 4.3.1

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3.2

Add

$3$ and

$4$ .

$\frac{\sqrt[6]{n^{7}}}{n}$

Step 5

Simplify the numerator.

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Step 5.1

Factor out

$n^{6}$ .

$\frac{\sqrt[6]{n^{6} n}}{n}$

Step 5.2

Pull terms out from under the radical.

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

Step 6

Cancel the common factor of

$n$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Divide

$\sqrt[6]{n}$ by

$1$ .

$\sqrt[6]{n}$