Perform the indicated operation, leaving all answers in simplest form. (5 points) 1. Simplify the following: sqrt(108) = sqrt(15) * sqrt(5) = 3 12/(2sqrt(3)) = 4. 4sqrt(3) + sqrt(27) 5. 3sqrt(12) - 2sqrt(3) = < # 5sqrt(3) 2sqrt(3) 7sqrt(3) pi * sqrt(3) :: sqrt(75) = 3sqrt(3) : 4sqrt(3) + sqrt(27) plus/minus 3 * sqrt(12) plus/minus 4 * sqrt(3) : 3sqrt(12) - 2sqrt(3) = 8sqrt(3) 1/6 * sqrt(3) :: (6sqrt(3))/3

Answer 1

Let's go through the operations step-by-step, simplifying each one as we go, ensuring that we leave answers in their simplest form.

1. Simplify \( \sqrt{108} \): \[ \sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \]

2. Calculate \( \sqrt{15} \cdot \sqrt{5} \): \[ \sqrt{15} \cdot \sqrt{5} = \sqrt{15 \cdot 5} = \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} \]

3. Simplify \( \frac{12}{2\sqrt{3}} \): \[ \frac{12}{2\sqrt{3}} = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} \]

4. Simplify \( 4\sqrt{3} + \sqrt{27} \): \[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \] Thus, \[ 4\sqrt{3} + \sqrt{27} = 4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3} \]

5. Calculate \( 3\sqrt{12} - 2\sqrt{3} \): \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \] Hence, \[ 3\sqrt{12} = 3 \cdot 2\sqrt{3} = 6\sqrt{3} \] Therefore, \[ 3\sqrt{12} - 2\sqrt{3} = 6\sqrt{3} - 2\sqrt{3} = 4\sqrt{3} \]

6. Simplify \( \sqrt{75} \): \[ \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} \]

7. Perform the calculation \( 4\sqrt{3} + \sqrt{27} + 3\sqrt{12} - 2\sqrt{3} \): From previous calculations: \[ 4\sqrt{3} + 3\sqrt{27} = 7\sqrt{3} \] and \[ 3\sqrt{12} - 2\sqrt{3} = 4\sqrt{3} \] Therefore: \[ 4\sqrt{3} + 3\sqrt{12} + \sqrt{27} - 2\sqrt{3} = 7\sqrt{3} + 4\sqrt{3} = 11\sqrt{3} \]

8. Simplify \( \frac{6\sqrt{3}}{3} \): \[ \frac{6\sqrt{3}}{3} = 2\sqrt{3} \]

Putting this information together, we have simplified various expressions and combined their results. The final answers are as follows:

1. \( \sqrt{108} = 6\sqrt{3} \)
2. \( \sqrt{15} \cdot \sqrt{5} = 5\sqrt{3} \)
3. \( \frac{12}{2\sqrt{3}} = 2\sqrt{3} \)
4. \( 4\sqrt{3} + \sqrt{27} = 7\sqrt{3} \)
5. \( 3\sqrt{12} - 2\sqrt{3} = 4\sqrt{3} \)
6. \( \sqrt{75} = 5\sqrt{3} \)
7. \( 4\sqrt{3} + 3\sqrt{12} + \sqrt{27} - 2\sqrt{3} = 11\sqrt{3} \)
8. \( \frac{6\sqrt{3}}{3} = 2\sqrt{3} \)