Simplify 4 sixth root of 3+2 fourth root of 32-3 sixth root of 192-2 sixth root of 192

$4 \sqrt[6]{3} + 2 \sqrt[4]{32} - 3 \sqrt[6]{192} - 2 \sqrt[6]{192}$

Simplify each term.

Rewrite $32$ as $2^{4} \cdot 2$ .

Factor $16$ out of $32$ .

$4 \sqrt[6]{3} + 2 \sqrt[4]{16 (2)} - 3 \sqrt[6]{192} - 2 \sqrt[6]{192}$

Rewrite $16$ as $2^{4}$ .

$4 \sqrt[6]{3} + 2 \sqrt[4]{2^{4} \cdot 2} - 3 \sqrt[6]{192} - 2 \sqrt[6]{192}$

Pull terms out from under the radical.

$4 \sqrt[6]{3} + 2 (2 \sqrt[4]{2}) - 3 \sqrt[6]{192} - 2 \sqrt[6]{192}$

Multiply $2$ by $2$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 3 \sqrt[6]{192} - 2 \sqrt[6]{192}$

Rewrite $192$ as $2^{6} \cdot 3$ .

Factor $64$ out of $192$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 3 \sqrt[6]{64 (3)} - 2 \sqrt[6]{192}$

Rewrite $64$ as $2^{6}$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 3 \sqrt[6]{2^{6} \cdot 3} - 2 \sqrt[6]{192}$

Pull terms out from under the radical.

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 3 (2 \sqrt[6]{3}) - 2 \sqrt[6]{192}$

Multiply $2$ by $- 3$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 6 \sqrt[6]{3} - 2 \sqrt[6]{192}$

Rewrite $192$ as $2^{6} \cdot 3$ .

Factor $64$ out of $192$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 6 \sqrt[6]{3} - 2 \sqrt[6]{64 (3)}$

Rewrite $64$ as $2^{6}$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 6 \sqrt[6]{3} - 2 \sqrt[6]{2^{6} \cdot 3}$

Pull terms out from under the radical.

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 6 \sqrt[6]{3} - 2 (2 \sqrt[6]{3})$

Multiply $2$ by $- 2$ .

$4 \sqrt[6]{3} + 4 \sqrt[4]{2} - 6 \sqrt[6]{3} - 4 \sqrt[6]{3}$

Simplify by adding terms.

Subtract $6 \sqrt[6]{3}$ from $4 \sqrt[6]{3}$ .

$- 2 \sqrt[6]{3} + 4 \sqrt[4]{2} - 4 \sqrt[6]{3}$

Subtract $4 \sqrt[6]{3}$ from $- 2 \sqrt[6]{3}$ .

$- 6 \sqrt[6]{3} + 4 \sqrt[4]{2}$

The result can be shown in multiple forms.

Exact Form:

$- 6 \sqrt[6]{3} + 4 \sqrt[4]{2}$

Decimal Form:

$- 2.44879327 \dots$