Simplify the Radical Expression (2i)^5*(i square root of 6)^2

${(2 i)}^{5} \cdot {(i \sqrt{6})}^{2}$

Apply the product rule to $2 i$ .

$2^{5} i^{5} \cdot {(i \sqrt{6})}^{2}$

Raise $2$ to the power of $5$ .

$32 i^{5} \cdot {(i \sqrt{6})}^{2}$

Factor out $i^{4}$ .

$32 (i^{4} i) \cdot {(i \sqrt{6})}^{2}$

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

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

$32 ({(i^{2})}^{2} i) \cdot {(i \sqrt{6})}^{2}$

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

$32 ({(- 1)}^{2} i) \cdot {(i \sqrt{6})}^{2}$

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

$32 (1 i) \cdot {(i \sqrt{6})}^{2}$

Multiply $i$ by $1$ .

$32 i \cdot {(i \sqrt{6})}^{2}$

Apply the product rule to $i \sqrt{6}$ .

$32 i \cdot (i^{2} {\sqrt{6}}^{2})$

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

$32 i \cdot (- 1 {\sqrt{6}}^{2})$

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

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

$32 i \cdot (- 1 {(6^{\frac{1}{2}})}^{2})$

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

$32 i \cdot (- 1 \cdot 6^{\frac{1}{2} \cdot 2})$

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

$32 i \cdot (- 1 \cdot 6^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$32 i \cdot (- 1 \cdot 6^{\frac{2}{2}})$

Rewrite the expression.

$32 i \cdot (- 1 \cdot 6^{1})$

Evaluate the exponent.

$32 i \cdot (- 1 \cdot 6)$

Multiply $- 1$ by $6$ .

$32 i \cdot - 6$

Multiply $- 6$ by $32$ .

$- 192 i$