Algebra Examples

$\sqrt{8} \div \sqrt[3]{4}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{8}}{\sqrt[3]{4}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{\sqrt[3]{4}}$

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Multiply $\frac{2 \sqrt{2}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

Multiply $\frac{2 \sqrt{2}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 4.1.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

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

Step 4.1.3

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

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

Step 4.1.4

Add $1$ and $2$ .

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

Step 4.1.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

Cancel the common factor of $3$ .

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Step 4.1.5.4.1

Cancel the common factor.

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

Step 4.1.5.4.2

Rewrite the expression.

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

Step 4.1.5.5

Evaluate the exponent.

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

Step 4.2

Cancel the common factor of $2$ and $4$ .

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Step 4.2.1

Factor $2$ out of $2 \sqrt{2} {\sqrt[3]{4}}^{2}$ .

$\frac{2 (\sqrt{2} {\sqrt[3]{4}}^{2})}{4}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $4$ .

$\frac{2 (\sqrt{2} {\sqrt[3]{4}}^{2})}{2 \cdot 2}$

Step 4.2.2.2

Cancel the common factor.

$\frac{2 (\sqrt{2} {\sqrt[3]{4}}^{2})}{2 \cdot 2}$

Step 4.2.2.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

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

Step 5.2

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

$\frac{\sqrt{2} \sqrt[3]{16}}{2}$

Step 5.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 5.3.1

Factor $8$ out of $16$ .

$\frac{\sqrt{2} \sqrt[3]{8 (2)}}{2}$

Step 5.3.2

Rewrite $8$ as $2^{3}$ .

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

Step 5.4

Pull terms out from under the radical.

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

Step 5.5

Combine exponents.

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Step 5.5.1

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

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Step 5.5.1.1

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

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

Step 5.5.1.2

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

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

Step 5.5.1.3

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

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

Step 5.5.1.4

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

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

Step 5.5.1.5

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

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

Step 5.5.1.6

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

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

Step 5.5.2

Combine using the product rule for radicals.

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

Step 5.5.3

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

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Step 5.5.3.1

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

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

Step 5.5.3.2

Add $3$ and $2$ .

$\frac{2 \sqrt[6]{2^{5}}}{2}$

Step 5.6

Raise $2$ to the power of $5$ .

$\frac{2 \sqrt[6]{32}}{2}$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$\frac{2 \sqrt[6]{32}}{2}$

Step 6.2

Divide $\sqrt[6]{32}$ by $1$ .

$\sqrt[6]{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{32}$

Decimal Form:

$1.78179743 \dots$