Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite the expression using the least common index of $6$ .

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Step 3.1.1

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

$\frac{3^{\frac{1}{3}} \sqrt{3}}{3}$

Step 3.1.2

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{2}{6}}$ .

$\frac{3^{\frac{2}{6}} \sqrt{3}}{3}$

Step 3.1.3

Rewrite $3^{\frac{2}{6}}$ as $\sqrt[6]{3^{2}}$ .

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

Step 3.1.4

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

$\frac{\sqrt[6]{3^{2}} \cdot 3^{\frac{1}{2}}}{3}$

Step 3.1.5

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{3}{6}}$ .

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

Step 3.1.6

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

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

Step 3.2

Combine using the product rule for radicals.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Add $2$ and $3$ .

$\frac{\sqrt[6]{3^{5}}}{3}$

Step 4

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

$\frac{\sqrt[6]{243}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[6]{243}}{3}$

Decimal Form:

$0.83268317 \dots$