Basic Math Examples

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

Step 1

Remove unnecessary parentheses.

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

Step 2

Combine exponents.

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Step 2.1

Multiply $4$ by $2$ .

$8 \sqrt{2} \sqrt[3]{4}$

Step 2.2

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

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Step 2.2.1

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

$8 (4^{\frac{1}{3}} \sqrt{2})$

Step 2.2.2

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

$8 (4^{\frac{2}{6}} \sqrt{2})$

Step 2.2.3

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

$8 (\sqrt[6]{4^{2}} \sqrt{2})$

Step 2.2.4

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

$8 (\sqrt[6]{4^{2}} \cdot 2^{\frac{1}{2}})$

Step 2.2.5

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

$8 (\sqrt[6]{4^{2}} \cdot 2^{\frac{3}{6}})$

Step 2.2.6

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

$8 (\sqrt[6]{4^{2}} \sqrt[6]{2^{3}})$

Step 2.3

Combine using the product rule for radicals.

$8 \sqrt[6]{4^{2} \cdot 2^{3}}$

Step 2.4

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

$8 \sqrt[6]{{(2^{2})}^{2} \cdot 2^{3}}$

Step 2.5

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

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Step 2.5.1

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

$8 \sqrt[6]{2^{2 \cdot 2} \cdot 2^{3}}$

Step 2.5.2

Multiply $2$ by $2$ .

$8 \sqrt[6]{2^{4} \cdot 2^{3}}$

Step 2.6

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

$8 \sqrt[6]{2^{4 + 3}}$

Step 2.7

Add $4$ and $3$ .

$8 \sqrt[6]{2^{7}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt[6]{2^{7}}$

Decimal Form:

$17.95939277 \dots$