Find the Exact Value 3 square root of 12+1/2 square root of 96- square root of 54

$3 \sqrt{12} + \frac{1}{2} \sqrt{96} - \sqrt{54}$

Step 1

Simplify each term.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$3 \sqrt{4 (3)} + \frac{1}{2} \sqrt{96} - \sqrt{54}$

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

$3 \sqrt{2^{2} \cdot 3} + \frac{1}{2} \sqrt{96} - \sqrt{54}$

Pull terms out from under the radical.

$3 (2 \sqrt{3}) + \frac{1}{2} \sqrt{96} - \sqrt{54}$

Multiply $2$ by $3$ .

$6 \sqrt{3} + \frac{1}{2} \sqrt{96} - \sqrt{54}$

Rewrite $96$ as $4^{2} \cdot 6$ .

Factor $16$ out of $96$ .

$6 \sqrt{3} + \frac{1}{2} \sqrt{16 (6)} - \sqrt{54}$

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

$6 \sqrt{3} + \frac{1}{2} \sqrt{4^{2} \cdot 6} - \sqrt{54}$

Pull terms out from under the radical.

$6 \sqrt{3} + \frac{1}{2} (4 \sqrt{6}) - \sqrt{54}$

Cancel the common factor of $2$ .

Factor $2$ out of $4 \sqrt{6}$ .

$6 \sqrt{3} + \frac{1}{2} (2 (2 \sqrt{6})) - \sqrt{54}$

Cancel the common factor.

$6 \sqrt{3} + \frac{1}{2} (2 (2 \sqrt{6})) - \sqrt{54}$

Rewrite the expression.

$6 \sqrt{3} + 2 \sqrt{6} - \sqrt{54}$

Rewrite $54$ as $3^{2} \cdot 6$ .

Factor $9$ out of $54$ .

$6 \sqrt{3} + 2 \sqrt{6} - \sqrt{9 (6)}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $3$ by $- 1$ .

$6 \sqrt{3} + 2 \sqrt{6} - 3 \sqrt{6}$

Step 2

Subtract $3 \sqrt{6}$ from $2 \sqrt{6}$ .

$6 \sqrt{3} - \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{3} - \sqrt{6}$

Decimal Form:

$7.94281510 \dots$