Basic Math Examples

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

Step 1

Combine $\frac{1}{2}$ and $a^{4}$ .

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

Step 2

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

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

Step 3

Combine.

$\frac{1 \sqrt[3]{a^{4}}}{2 \sqrt[3]{2}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 4

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

$\frac{\sqrt[3]{a^{4}}}{2 \sqrt[3]{2}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 5

Simplify the numerator.

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Step 5.1

Factor out $a^{3}$ .

$\frac{\sqrt[3]{a^{3} a}}{2 \sqrt[3]{2}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 5.2

Pull terms out from under the radical.

$\frac{a \sqrt[3]{a}}{2 \sqrt[3]{2}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

$\frac{a \sqrt[3]{a} {\sqrt[3]{2}}^{2}}{2 \sqrt[3]{2} {\sqrt[3]{2}}^{2}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 7.2

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

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

Step 7.3

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 7.4

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

$\frac{a \sqrt[3]{a} {\sqrt[3]{2}}^{2}}{2 {\sqrt[3]{2}}^{1 + 2}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 7.5

Add $1$ and $2$ .

$\frac{a \sqrt[3]{a} {\sqrt[3]{2}}^{2}}{2 {\sqrt[3]{2}}^{3}} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $3$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Combine using the product rule for radicals.

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

Step 9

Multiply $2$ by $2$ .

$\frac{a \sqrt[3]{4 a}}{4} + \sqrt[3]{\frac{1}{8} a^{4}}$

Step 10

Combine $\frac{1}{8}$ and $a^{4}$ .

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

Step 11

Rewrite $\frac{a^{4}}{8}$ as ${(\frac{a}{2})}^{3} a$ .

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Step 11.1

Factor the perfect power $a^{3}$ out of $a^{4}$ .

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

Step 11.2

Factor the perfect power $2^{3}$ out of $8$ .

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

Step 11.3

Rearrange the fraction $\frac{a^{3} a}{2^{3} \cdot 1}$ .

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

Step 12

Pull terms out from under the radical.

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

Step 13

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

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