Simplify the Radical Expression -3 sixth root of 3-2 sixth root of 192- sixth root of 320

$- 3 \sqrt[6]{3} - 2 \sqrt[6]{192} - \sqrt[6]{320}$

Step 1

Simplify each term.

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

Factor $64$ out of $192$ .

$- 3 \sqrt[6]{3} - 2 \sqrt[6]{64 (3)} - \sqrt[6]{320}$

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

$- 3 \sqrt[6]{3} - 2 \sqrt[6]{2^{6} \cdot 3} - \sqrt[6]{320}$

Pull terms out from under the radical.

$- 3 \sqrt[6]{3} - 2 (| 2 | \sqrt[6]{3}) - \sqrt[6]{320}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$- 3 \sqrt[6]{3} - 2 (2 \sqrt[6]{3}) - \sqrt[6]{320}$

Multiply $2$ by $- 2$ .

$- 3 \sqrt[6]{3} - 4 \sqrt[6]{3} - \sqrt[6]{320}$

Rewrite $320$ as $2^{6} \cdot 5$ .

Factor $64$ out of $320$ .

$- 3 \sqrt[6]{3} - 4 \sqrt[6]{3} - \sqrt[6]{64 (5)}$

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

$- 3 \sqrt[6]{3} - 4 \sqrt[6]{3} - \sqrt[6]{2^{6} \cdot 5}$

Pull terms out from under the radical.

$- 3 \sqrt[6]{3} - 4 \sqrt[6]{3} - (| 2 | \sqrt[6]{5})$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$- 3 \sqrt[6]{3} - 4 \sqrt[6]{3} - (2 \sqrt[6]{5})$

Multiply $2$ by $- 1$ .

$- 3 \sqrt[6]{3} - 4 \sqrt[6]{3} - 2 \sqrt[6]{5}$

Step 2

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

$- 7 \sqrt[6]{3} - 2 \sqrt[6]{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 7 \sqrt[6]{3} - 2 \sqrt[6]{5}$

Decimal Form:

$- 11.02187965 \dots$