Pre-Algebra Examples

$\frac{\sqrt[3]{2 \sqrt{10}}}{\sqrt{10 \sqrt[3]{2}}}$

Step 1

Multiply $\frac{\sqrt[3]{2 \sqrt{10}}}{\sqrt{10 \sqrt[3]{2}}}$ by $\frac{\sqrt{10 \sqrt[3]{2}}}{\sqrt{10 \sqrt[3]{2}}}$ .

$\frac{\sqrt[3]{2 \sqrt{10}}}{\sqrt{10 \sqrt[3]{2}}} \cdot \frac{\sqrt{10 \sqrt[3]{2}}}{\sqrt{10 \sqrt[3]{2}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[3]{2 \sqrt{10}}}{\sqrt{10 \sqrt[3]{2}}}$ by $\frac{\sqrt{10 \sqrt[3]{2}}}{\sqrt{10 \sqrt[3]{2}}}$ .

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{\sqrt{10 \sqrt[3]{2}} \sqrt{10 \sqrt[3]{2}}}$

Step 2.2

Raise $\sqrt{10 \sqrt[3]{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{\sqrt{10 \sqrt[3]{2}}}^{1} \sqrt{10 \sqrt[3]{2}}}$

Step 2.3

Raise $\sqrt{10 \sqrt[3]{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{\sqrt{10 \sqrt[3]{2}}}^{1} {\sqrt{10 \sqrt[3]{2}}}^{1}}$

Step 2.4

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

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{\sqrt{10 \sqrt[3]{2}}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{\sqrt{10 \sqrt[3]{2}}}^{2}}$

Step 2.6

Rewrite ${\sqrt{10 \sqrt[3]{2}}}^{2}$ as $10 \sqrt[3]{2}$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 \sqrt[3]{2}}$ as ${(10 \sqrt[3]{2})}^{\frac{1}{2}}$ .

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{({(10 \sqrt[3]{2})}^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{(10 \sqrt[3]{2})}^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{(10 \sqrt[3]{2})}^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{(10 \sqrt[3]{2})}^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{{(10 \sqrt[3]{2})}^{1}}$

Step 2.6.5

Simplify.

$\frac{\sqrt[3]{2 \sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{10 \sqrt[3]{2}}$

Step 3

Combine $\sqrt[3]{2 \sqrt{10}}$ and $\sqrt[3]{2}$ into a single radical.

$\frac{\sqrt[3]{\frac{2 \sqrt{10}}{2}} \sqrt{10 \sqrt[3]{2}}}{10}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$\frac{\sqrt[3]{\frac{2 \sqrt{10}}{2}} \sqrt{10 \sqrt[3]{2}}}{10}$

Step 4.2

Divide $\sqrt{10}$ by $1$ .

$\frac{\sqrt[3]{\sqrt{10}} \sqrt{10 \sqrt[3]{2}}}{10}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\sqrt{10}}$ as ${\sqrt{10}}^{\frac{1}{3}}$ .

$\frac{{\sqrt{10}}^{\frac{1}{3}} \sqrt{10 \sqrt[3]{2}}}{10}$

Step 5.1.2

Rewrite ${\sqrt{10}}^{\frac{1}{3}}$ as ${\sqrt{10}}^{\frac{2}{6}}$ .

$\frac{{\sqrt{10}}^{\frac{2}{6}} \sqrt{10 \sqrt[3]{2}}}{10}$

Step 5.1.3

Rewrite ${\sqrt{10}}^{\frac{2}{6}}$ as $\sqrt[6]{{\sqrt{10}}^{2}}$ .

$\frac{\sqrt[6]{{\sqrt{10}}^{2}} \sqrt{10 \sqrt[3]{2}}}{10}$

Step 5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 \sqrt[3]{2}}$ as ${(10 \sqrt[3]{2})}^{\frac{1}{2}}$ .

$\frac{\sqrt[6]{{\sqrt{10}}^{2}} {(10 \sqrt[3]{2})}^{\frac{1}{2}}}{10}$

Step 5.1.5

Rewrite ${(10 \sqrt[3]{2})}^{\frac{1}{2}}$ as ${(10 \sqrt[3]{2})}^{\frac{3}{6}}$ .

$\frac{\sqrt[6]{{\sqrt{10}}^{2}} {(10 \sqrt[3]{2})}^{\frac{3}{6}}}{10}$

Step 5.1.6

Rewrite ${(10 \sqrt[3]{2})}^{\frac{3}{6}}$ as $\sqrt[6]{{(10 \sqrt[3]{2})}^{3}}$ .

$\frac{\sqrt[6]{{\sqrt{10}}^{2}} \sqrt[6]{{(10 \sqrt[3]{2})}^{3}}}{10}$

Step 5.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{{\sqrt{10}}^{2} {(10 \sqrt[3]{2})}^{3}}}{10}$

Step 5.3

Apply the product rule to $10 \sqrt[3]{2}$ .

$\frac{\sqrt[6]{{\sqrt{10}}^{2} \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.4

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt[6]{{(10^{\frac{1}{2}})}^{2} \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.4.2

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

$\frac{\sqrt[6]{10^{\frac{1}{2} \cdot 2} \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.4.3

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

$\frac{\sqrt[6]{10^{\frac{2}{2}} \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.4.4

Cancel the common factor of $2$ .

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Step 5.4.4.1

Cancel the common factor.

$\frac{\sqrt[6]{10^{\frac{2}{2}} \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.4.4.2

Rewrite the expression.

$\frac{\sqrt[6]{10^{1} \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.4.5

Evaluate the exponent.

$\frac{\sqrt[6]{10 \cdot 10^{3} {\sqrt[3]{2}}^{3}}}{10}$

Step 5.5

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

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Step 5.5.1

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

$\frac{\sqrt[6]{10 \cdot 10^{3} {(2^{\frac{1}{3}})}^{3}}}{10}$

Step 5.5.2

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

$\frac{\sqrt[6]{10 \cdot 10^{3} \cdot 2^{\frac{1}{3} \cdot 3}}}{10}$

Step 5.5.3

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

$\frac{\sqrt[6]{10 \cdot 10^{3} \cdot 2^{\frac{3}{3}}}}{10}$

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{\sqrt[6]{10 \cdot 10^{3} \cdot 2^{\frac{3}{3}}}}{10}$

Step 5.5.4.2

Rewrite the expression.

$\frac{\sqrt[6]{10 \cdot 10^{3} \cdot 2^{1}}}{10}$

Step 5.5.5

Evaluate the exponent.

$\frac{\sqrt[6]{10 \cdot 10^{3} \cdot 2}}{10}$

Step 5.6

Multiply $2$ by $10$ .

$\frac{\sqrt[6]{20 \cdot 10^{3}}}{10}$

Step 5.7

Raise $10$ to the power of $3$ .

$\frac{\sqrt[6]{20 \cdot 1000}}{10}$

Step 6

Multiply $20$ by $1000$ .

$\frac{\sqrt[6]{20000}}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[6]{20000}}{10}$

Decimal Form:

$0.52100073 \dots$