Question

what is $2(12)^{\frac{3}{2}}$ in its simplest form?
$48\sqrt{3}$
432
36
$48\sqrt{6}$

Explanation:

Step1: Rewrite exponent as root

Recall that $a^{\frac{3}{2}} = (\sqrt{a})^3$. So:
$2(12)^{\frac{3}{2}} = 2 \times (\sqrt{12})^3$

Step2: Simplify $\sqrt{12}$

Factor 12 into $4 \times 3$, so $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$. Substitute back:
$2 \times (2\sqrt{3})^3$

Step3: Calculate $(2\sqrt{3})^3$

Expand the power: $(2\sqrt{3})^3 = 2^3 \times (\sqrt{3})^3 = 8 \times 3\sqrt{3} = 24\sqrt{3}$

Step4: Multiply by the leading coefficient

$2 \times 24\sqrt{3} = 48\sqrt{3}$

Answer:

$48\sqrt{3}$