Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Evaluate the exponent.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

$\frac{\sqrt{10} \sqrt[3]{4}}{2}$

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

$\frac{10^{\frac{1}{2}} \sqrt[3]{4}}{2}$

Step 5.1.2

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

$\frac{10^{\frac{3}{6}} \sqrt[3]{4}}{2}$

Step 5.1.3

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

$\frac{\sqrt[6]{10^{3}} \sqrt[3]{4}}{2}$

Step 5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{\sqrt[6]{10^{3}} \cdot 4^{\frac{1}{3}}}{2}$

Step 5.1.5

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$\frac{\sqrt[6]{10^{3}} \cdot 4^{\frac{2}{6}}}{2}$

Step 5.1.6

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$\frac{\sqrt[6]{10^{3}} \sqrt[6]{4^{2}}}{2}$

Step 5.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{10^{3} \cdot 4^{2}}}{2}$

Step 5.3

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

$\frac{\sqrt[6]{1000 \cdot 4^{2}}}{2}$

Step 5.4

Raise $4$ to the power of $2$ .

$\frac{\sqrt[6]{1000 \cdot 16}}{2}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $1000$ by $16$ .

$\frac{\sqrt[6]{16000}}{2}$

Step 6.2

Rewrite $16000$ as $2^{6} \cdot 250$ .

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Step 6.2.1

Factor $64$ out of $16000$ .

$\frac{\sqrt[6]{64 (250)}}{2}$

Step 6.2.2

Rewrite $64$ as $2^{6}$ .

$\frac{\sqrt[6]{2^{6} \cdot 250}}{2}$

Step 6.3

Pull terms out from under the radical.

$\frac{| 2 | \sqrt[6]{250}}{2}$

Step 6.4

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 \sqrt[6]{250}}{2}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$\frac{2 \sqrt[6]{250}}{2}$

Step 7.2

Divide $\sqrt[6]{250}$ by $1$ .

$\sqrt[6]{250}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{250}$

Decimal Form:

$2.50990144 \dots$