Basic Math Examples

${(\sqrt[3]{2} \cdot \sqrt[4]{4})}^{3} \div \sqrt[6]{2} \div \sqrt[3]{4}$

Step 1

Rewrite the division as a fraction.

$\frac{{(\sqrt[3]{2} \cdot \sqrt[4]{4})}^{3} \div \sqrt[6]{2}}{\sqrt[3]{4}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

${(\sqrt[3]{2} \cdot \sqrt[4]{4})}^{3} \div \sqrt[6]{2} \cdot \frac{1}{\sqrt[3]{4}}$

Step 3

Combine fractions.

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Step 3.1

Combine.

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

Step 3.2

Multiply ${(\sqrt[3]{2} \cdot \sqrt[4]{4})}^{3}$ by $1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

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

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

Step 4.4

Multiply the exponents in ${(2^{2})}^{2}$ .

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Step 4.4.1

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

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

Step 4.4.2

Multiply $2$ by $2$ .

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

Step 4.5

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

$\frac{{(\sqrt[3]{2} \cdot \sqrt[4]{4})}^{3}}{\sqrt[6]{2^{1 + 4}}}$

Step 4.6

Add $1$ and $4$ .

$\frac{{(\sqrt[3]{2} \cdot \sqrt[4]{4})}^{3}}{\sqrt[6]{2^{5}}}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the product rule to $\sqrt[3]{2} \cdot \sqrt[4]{4}$ .

$\frac{{\sqrt[3]{2}}^{3} \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{{(2^{\frac{1}{3}})}^{3} \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.2.2

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

$\frac{2^{\frac{1}{3} \cdot 3} \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.2.3

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

$\frac{2^{\frac{3}{3}} \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.2.4

Cancel the common factor of $3$ .

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Step 5.2.4.1

Cancel the common factor.

$\frac{2^{\frac{3}{3}} \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.2.4.2

Rewrite the expression.

$\frac{2^{1} \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.2.5

Evaluate the exponent.

$\frac{2 \cdot {\sqrt[4]{4}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.3

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

$\frac{2 \cdot {\sqrt[4]{2^{2}}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.4

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

$\frac{2 \cdot {\sqrt{\sqrt{2^{2}}}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.5

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

$\frac{2 \cdot {\sqrt{2}}^{3}}{\sqrt[6]{2^{5}}}$

Step 5.6

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

$\frac{2 \cdot \sqrt{2^{3}}}{\sqrt[6]{2^{5}}}$

Step 5.7

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

$\frac{2 \cdot \sqrt{8}}{\sqrt[6]{2^{5}}}$

Step 5.8

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

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Step 5.8.1

Factor $4$ out of $8$ .

$\frac{2 \cdot \sqrt{4 (2)}}{\sqrt[6]{2^{5}}}$

Step 5.8.2

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

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

Step 5.9

Pull terms out from under the radical.

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

Step 5.10

Multiply $2$ by $2$ .

$\frac{4 \sqrt{2}}{\sqrt[6]{2^{5}}}$

Step 6

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

$\frac{4 \sqrt{2}}{\sqrt[6]{32}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \sqrt{2}}{\sqrt[6]{32}}$

Decimal Form:

$3.17480210 \dots$