Simplify the Radical Expression (4 square root of 3a^3)^2

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

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

Factor out $a^{2}$ .

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

Reorder $3$ and $a^{2}$ .

${(4 \sqrt{a^{2} \cdot 3 a})}^{2}$

Add parentheses.

${(4 \sqrt{a^{2} \cdot (3 a)})}^{2}$

Pull terms out from under the radical.

${(4 (| a | \sqrt{3 a}))}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $4 | a | \sqrt{3 a}$ .

${(4 | a |)}^{2} {\sqrt{3 a}}^{2}$

Apply the product rule to $4 | a |$ .

$4^{2} {| a |}^{2} {\sqrt{3 a}}^{2}$

Raise $4$ to the power of $2$ .

$16 {| a |}^{2} {\sqrt{3 a}}^{2}$

Remove the absolute value in ${| a |}^{2}$ because exponentiations with even powers are always positive.

$16 a^{2} {\sqrt{3 a}}^{2}$

Rewrite ${\sqrt{3 a}}^{2}$ as $3 a$ .

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

$16 a^{2} {({(3 a)}^{\frac{1}{2}})}^{2}$

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

$16 a^{2} {(3 a)}^{\frac{1}{2} \cdot 2}$

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

$16 a^{2} {(3 a)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 a^{2} {(3 a)}^{\frac{2}{2}}$

Rewrite the expression.

$16 a^{2} {(3 a)}^{1}$

Simplify.

$16 a^{2} (3 a)$

Rewrite using the commutative property of multiplication.

$16 \cdot 3 a^{2} a$

Multiply $a^{2}$ by $a$ by adding the exponents.

Move $a$ .

$16 \cdot 3 (a \cdot a^{2})$

Multiply $a$ by $a^{2}$ .

Raise $a$ to the power of $1$ .

$16 \cdot 3 (a^{1} a^{2})$

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

$16 \cdot 3 a^{1 + 2}$

Add $1$ and $2$ .

$16 \cdot 3 a^{3}$

Multiply $16$ by $3$ .

$48 a^{3}$

Simplify the Radical Expression (4 square root of 3a^3)^2

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