Simplify fourth root of 16(x-3)^12

$\sqrt[4]{16 {(x - 3)}^{12}}$

Rewrite $16 {(x - 3)}^{12}$ as ${(2 {(x - 3)}^{3})}^{4}$ .

$\sqrt[4]{{(2 {(x - 3)}^{3})}^{4}}$

Pull terms out from under the radical, assuming positive real numbers.

$2 {(x - 3)}^{3}$

Use the Binomial Theorem.

$2 (x^{3} + 3 x^{2} \cdot - 3 + 3 x {(- 3)}^{2} + {(- 3)}^{3})$

Simplify terms.

Simplify each term.

Multiply $- 3$ by $3$ .

$2 (x^{3} - 9 x^{2} + 3 x {(- 3)}^{2} + {(- 3)}^{3})$

Raise $- 3$ to the power of $2$ .

$2 (x^{3} - 9 x^{2} + 3 x \cdot 9 + {(- 3)}^{3})$

Multiply $9$ by $3$ .

$2 (x^{3} - 9 x^{2} + 27 x + {(- 3)}^{3})$

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

$2 (x^{3} - 9 x^{2} + 27 x - 27)$

Apply the distributive property.

$2 x^{3} + 2 (- 9 x^{2}) + 2 (27 x) + 2 \cdot - 27$

Simplify.

Multiply $- 9$ by $2$ .

$2 x^{3} - 18 x^{2} + 2 (27 x) + 2 \cdot - 27$

Multiply $27$ by $2$ .

$2 x^{3} - 18 x^{2} + 54 x + 2 \cdot - 27$

Multiply $2$ by $- 27$ .

$2 x^{3} - 18 x^{2} + 54 x - 54$