Basic Math Examples

$(2 \sqrt[3]{24} + \frac{1}{4} \cdot \sqrt[3]{3} + 2 \sqrt[3]{3}) \sqrt{2 a}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $24$ as $2^{3} \cdot 3$ .

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Step 1.1.1

Factor $8$ out of $24$ .

$(2 \sqrt[3]{8 (3)} + \frac{1}{4} \cdot \sqrt[3]{3} + 2 \sqrt[3]{3}) \sqrt{2 a}$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $2$ .

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

Step 1.4

Combine $\frac{1}{4}$ and $\sqrt[3]{3}$ .

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

Step 2

Find the common denominator.

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Step 2.1

Write $4 \sqrt[3]{3}$ as a fraction with denominator $1$ .

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

Step 2.2

Multiply $\frac{4 \sqrt[3]{3}}{1}$ by $\frac{4}{4}$ .

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

Step 2.3

Multiply $\frac{4 \sqrt[3]{3}}{1}$ by $\frac{4}{4}$ .

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

Step 2.4

Write $2 \sqrt[3]{3}$ as a fraction with denominator $1$ .

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

Step 2.5

Multiply $\frac{2 \sqrt[3]{3}}{1}$ by $\frac{4}{4}$ .

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

Step 2.6

Multiply $\frac{2 \sqrt[3]{3}}{1}$ by $\frac{4}{4}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $4$ by $4$ .

$\frac{16 \sqrt[3]{3} + \sqrt[3]{3} + 2 \sqrt[3]{3} \cdot 4}{4} \sqrt{2 a}$

Step 4.2

Multiply $4$ by $2$ .

$\frac{16 \sqrt[3]{3} + \sqrt[3]{3} + 8 \sqrt[3]{3}}{4} \sqrt{2 a}$

Step 5

Simplify by adding terms.

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Step 5.1

Add $16 \sqrt[3]{3}$ and $\sqrt[3]{3}$ .

$\frac{17 \sqrt[3]{3} + 8 \sqrt[3]{3}}{4} \sqrt{2 a}$

Step 5.2

Add $17 \sqrt[3]{3}$ and $8 \sqrt[3]{3}$ .

$\frac{25 \sqrt[3]{3}}{4} \sqrt{2 a}$

Step 6

Multiply $\frac{25 \sqrt[3]{3}}{4} \sqrt{2 a}$ .

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Step 6.1

Combine $\frac{25 \sqrt[3]{3}}{4}$ and $\sqrt{2 a}$ .

$\frac{25 \sqrt[3]{3} \sqrt{2 a}}{4}$

Step 6.2

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

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Step 6.2.1

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

$\frac{25 ({(2 a)}^{\frac{1}{2}} \sqrt[3]{3})}{4}$

Step 6.2.2

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

$\frac{25 ({(2 a)}^{\frac{3}{6}} \sqrt[3]{3})}{4}$

Step 6.2.3

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

$\frac{25 (\sqrt[6]{{(2 a)}^{3}} \sqrt[3]{3})}{4}$

Step 6.2.4

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

$\frac{25 (\sqrt[6]{{(2 a)}^{3}} \cdot 3^{\frac{1}{3}})}{4}$

Step 6.2.5

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

$\frac{25 (\sqrt[6]{{(2 a)}^{3}} \cdot 3^{\frac{2}{6}})}{4}$

Step 6.2.6

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

$\frac{25 (\sqrt[6]{{(2 a)}^{3}} \sqrt[6]{3^{2}})}{4}$

Step 6.3

Combine using the product rule for radicals.

$\frac{25 \sqrt[6]{{(2 a)}^{3} \cdot 3^{2}}}{4}$

Step 6.4

Apply the product rule to $2 a$ .

$\frac{25 \sqrt[6]{2^{3} a^{3} \cdot 3^{2}}}{4}$

Step 6.5

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

$\frac{25 \sqrt[6]{8 a^{3} \cdot 3^{2}}}{4}$

Step 6.6

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

$\frac{25 \sqrt[6]{8 a^{3} \cdot 9}}{4}$

Step 6.7

Multiply $9$ by $8$ .

$\frac{25 \sqrt[6]{72 a^{3}}}{4}$