Basic Math Examples

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

Step 1

Multiply $\sqrt{2} \sqrt[3]{2}$ .

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.2

Combine using the product rule for radicals.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Add $3$ and $2$ .

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

Step 2

Combine using the product rule for radicals.

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

Step 3

Multiply $2^{5}$ by $2$ by adding the exponents.

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Step 3.1

Multiply $2^{5}$ by $2$ .

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Step 3.1.1

Raise $2$ to the power of $1$ .

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

Step 3.1.2

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

$\sqrt[6]{2^{5 + 1}}$

Step 3.2

Add $5$ and $1$ .

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

Step 4

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

$2$