Basic Math Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Evaluate the exponent.

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

Step 3

Cancel the common factor of $3$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

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

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

Step 4

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

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

Step 5

Raise $3$ to the power of $2$ .

$\sqrt{\sqrt[3]{9}}$

Step 6

Rewrite $\sqrt{\sqrt[3]{9}}$ as $\sqrt[6]{9}$ .

$\sqrt[6]{9}$

Step 7

Rewrite $9$ as $3^{2}$ .

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

Step 8

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

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

Step 9

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[3]{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\sqrt[3]{3}$

Decimal Form:

$1.44224957 \dots$