Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

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

Step 1.1.3

Factor out $a^{2}$ .

$\sqrt{2 a} + 2 \sqrt{2^{2} \cdot 2 (a^{2} a)} + 2 a \sqrt{\frac{1}{2} a}$

Step 1.1.4

Move $2$ .

$\sqrt{2 a} + 2 \sqrt{2^{2} a^{2} \cdot 2 a} + 2 a \sqrt{\frac{1}{2} a}$

Step 1.1.5

Rewrite $2^{2} a^{2}$ as ${(2 a)}^{2}$ .

$\sqrt{2 a} + 2 \sqrt{{(2 a)}^{2} \cdot 2 a} + 2 a \sqrt{\frac{1}{2} a}$

Step 1.1.6

Add parentheses.

$\sqrt{2 a} + 2 \sqrt{{(2 a)}^{2} \cdot (2 a)} + 2 a \sqrt{\frac{1}{2} a}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{2 a} + 2 (2 a \sqrt{2 a}) + 2 a \sqrt{\frac{1}{2} a}$

Step 1.3

Multiply $2$ by $2$ .

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

Step 1.4

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

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \sqrt{\frac{a}{2}}$

Step 1.5

Rewrite $\sqrt{\frac{a}{2}}$ as $\frac{\sqrt{a}}{\sqrt{2}}$ .

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a}}{\sqrt{2}}$

Step 1.6

Multiply $\frac{\sqrt{a}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply $\frac{\sqrt{a}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.7.2

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

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.7.3

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

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.7.4

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

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.7.5

Add $1$ and $1$ .

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.7.6

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

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Step 1.7.6.1

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

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.7.6.2

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

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

Step 1.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{2^{1}}$

Step 1.7.6.5

Evaluate the exponent.

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a} \sqrt{2}}{2}$

Step 1.8

Combine using the product rule for radicals.

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a \cdot 2}}{2}$

Step 1.9

Cancel the common factor of $2$ .

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Step 1.9.1

Factor $2$ out of $2 a$ .

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

Step 1.9.2

Cancel the common factor.

$\sqrt{2 a} + 4 a \sqrt{2 a} + 2 a \frac{\sqrt{a \cdot 2}}{2}$

Step 1.9.3

Rewrite the expression.

$\sqrt{2 a} + 4 a \sqrt{2 a} + a \sqrt{a \cdot 2}$

Step 2

Add $4 a \sqrt{2 a}$ and $a \sqrt{a \cdot 2}$ .

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Step 2.1

Reorder $a$ and $2$ .

$\sqrt{2 a} + 4 a \sqrt{2 a} + a \sqrt{2 \cdot a}$

Step 2.2

Add $4 a \sqrt{2 a}$ and $a \sqrt{2 \cdot a}$ .

$\sqrt{2 a} + 5 a \sqrt{2 a}$