Simplify the Radical Expression square root of 2a+2 square root of 8a^3+2a square root of 1/2a

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

Simplify each term.

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

Factor $4$ out of $8$ .

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

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

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

Factor out $a^{2}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Combine using the product rule for radicals.

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 a$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Reorder factors in $\sqrt{2 a} + 4 | a | \sqrt{2 a} + a \sqrt{a \cdot 2}$ .

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

Simplify the Radical Expression square root of 2a+2 square root of 8a^3+2a square root of 1/2a

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