Evaluate fourth root of 256(5x-2)^12

$\sqrt[4]{256 {(5 x - 2)}^{12}}$

Rewrite $256 {(5 x - 2)}^{12}$ as ${(4 {(5 x - 2)}^{3})}^{4}$ .

$\sqrt[4]{{(4 {(5 x - 2)}^{3})}^{4}}$

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

$4 {(5 x - 2)}^{3}$

Use the Binomial Theorem.

$4 ({(5 x)}^{3} + 3 {(5 x)}^{2} \cdot - 2 + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

Simplify terms.

Simplify each term.

Apply the product rule to $5 x$ .

$4 (5^{3} x^{3} + 3 {(5 x)}^{2} \cdot - 2 + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

Raise $5$ to the power of $3$ .

$4 (125 x^{3} + 3 {(5 x)}^{2} \cdot - 2 + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

Apply the product rule to $5 x$ .

$4 (125 x^{3} + 3 (5^{2} x^{2}) \cdot - 2 + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

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

$4 (125 x^{3} + 3 (25 x^{2}) \cdot - 2 + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

Multiply $25$ by $3$ .

$4 (125 x^{3} + 75 x^{2} \cdot - 2 + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

Multiply $- 2$ by $75$ .

$4 (125 x^{3} - 150 x^{2} + 3 (5 x) {(- 2)}^{2} + {(- 2)}^{3})$

Multiply $5$ by $3$ .

$4 (125 x^{3} - 150 x^{2} + 15 x {(- 2)}^{2} + {(- 2)}^{3})$

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

$4 (125 x^{3} - 150 x^{2} + 15 x \cdot 4 + {(- 2)}^{3})$

Multiply $4$ by $15$ .

$4 (125 x^{3} - 150 x^{2} + 60 x + {(- 2)}^{3})$

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

$4 (125 x^{3} - 150 x^{2} + 60 x - 8)$

Apply the distributive property.

$4 (125 x^{3}) + 4 (- 150 x^{2}) + 4 (60 x) + 4 \cdot - 8$

Simplify.

Multiply $125$ by $4$ .

$500 x^{3} + 4 (- 150 x^{2}) + 4 (60 x) + 4 \cdot - 8$

Multiply $- 150$ by $4$ .

$500 x^{3} - 600 x^{2} + 4 (60 x) + 4 \cdot - 8$

Multiply $60$ by $4$ .

$500 x^{3} - 600 x^{2} + 240 x + 4 \cdot - 8$

Multiply $4$ by $- 8$ .

$500 x^{3} - 600 x^{2} + 240 x - 32$