Simplify (i^4 square root of 6)^2*(-4i)^3

${(i^{4} \sqrt{6})}^{2} \cdot {(- 4 i)}^{3}$

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

${({(i^{2})}^{2} \sqrt{6})}^{2} \cdot {(- 4 i)}^{3}$

Rewrite $i^{2}$ as $- 1$ .

${({(- 1)}^{2} \sqrt{6})}^{2} \cdot {(- 4 i)}^{3}$

Raise $- 1$ to the power of $2$ .

${(1 \sqrt{6})}^{2} \cdot {(- 4 i)}^{3}$

Multiply $\sqrt{6}$ by $1$ .

${\sqrt{6}}^{2} \cdot {(- 4 i)}^{3}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

${(6^{\frac{1}{2}})}^{2} \cdot {(- 4 i)}^{3}$

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

$6^{\frac{1}{2} \cdot 2} \cdot {(- 4 i)}^{3}$

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

$6^{\frac{2}{2}} \cdot {(- 4 i)}^{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$6^{\frac{2}{2}} \cdot {(- 4 i)}^{3}$

Rewrite the expression.

$6^{1} \cdot {(- 4 i)}^{3}$

Evaluate the exponent.

$6 \cdot {(- 4 i)}^{3}$

Simplify the expression.

Apply the product rule to $- 4 i$ .

$6 \cdot ({(- 4)}^{3} i^{3})$

Raise $- 4$ to the power of $3$ .

$6 \cdot (- 64 i^{3})$

Factor out $i^{2}$ .

$6 \cdot (- 64 (i^{2} \cdot i))$

Rewrite $i^{2}$ as $- 1$ .

$6 \cdot (- 64 (- 1 \cdot i))$

Rewrite $- 1 i$ as $- i$ .

$6 \cdot (- 64 (- i))$

Multiply $- 1$ by $- 64$ .

$6 \cdot (64 i)$

Multiply $64$ by $6$ .

$384 i$