Simplify the Radical Expression 6 square root of 2-5 square root of 8+2 square root of 32

$6 \sqrt{2} - 5 \sqrt{8} + 2 \sqrt{32}$

Step 1

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$6 \sqrt{2} - 5 \sqrt{4 (2)} + 2 \sqrt{32}$

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

$6 \sqrt{2} - 5 \sqrt{2^{2} \cdot 2} + 2 \sqrt{32}$

Pull terms out from under the radical.

$6 \sqrt{2} - 5 (| 2 | \sqrt{2}) + 2 \sqrt{32}$

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

$6 \sqrt{2} - 5 (2 \sqrt{2}) + 2 \sqrt{32}$

Multiply $2$ by $- 5$ .

$6 \sqrt{2} - 10 \sqrt{2} + 2 \sqrt{32}$

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

Factor $16$ out of $32$ .

$6 \sqrt{2} - 10 \sqrt{2} + 2 \sqrt{16 (2)}$

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

$6 \sqrt{2} - 10 \sqrt{2} + 2 \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$6 \sqrt{2} - 10 \sqrt{2} + 2 (| 4 | \sqrt{2})$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$6 \sqrt{2} - 10 \sqrt{2} + 2 (4 \sqrt{2})$

Multiply $4$ by $2$ .

$6 \sqrt{2} - 10 \sqrt{2} + 8 \sqrt{2}$

Step 2

Subtract $10 \sqrt{2}$ from $6 \sqrt{2}$ .

$- 4 \sqrt{2} + 8 \sqrt{2}$

Step 3

Add $- 4 \sqrt{2}$ and $8 \sqrt{2}$ .

$4 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{2}$

Decimal Form:

$5.65685424 \dots$