Question

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. \( \frac{6\sqrt{3}}{3} = 2\sqrt{3} \)