Question

Perform the indicated operation, leaving all answers in simplest form.(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Simplify the following:
108
−
−
−
√
108=
Response area


15
−
−
√
⋅
5
–
√
15⋅5=
Response area


12
2
3
√
1223=
Response area


4
3
–
√
+
27
−
−
√
43+27=
Response area


3
12
−
−
√
−2
3
–
√
312−23=
Response area


6
3
–
√
63
2
3
–
√
23
7
3
–
√
73
4
3
–
√
+
27
−
−
√
43+27
75
−
−
√
75
8
3
–
√
83
3
12
−
−
√
312
3
12
−
−
√
−2
3
–
√
312−23
4
3
–
√
43
3
3
–
√
33
3
–
√
3
5
3
–
√
53
6
3
√
3
633

Answer 1

Let's simplify each expression one by one:

1. **Simplifying \( \sqrt{108} \):**
\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}
\]
**Response:** \( 6\sqrt{3} \)

2. **Simplifying \( \sqrt{15} \cdot 5 \):**
\[
\sqrt{15} \cdot 5 = 5\sqrt{15}
\]
**Response:** \( 5\sqrt{15} \)

3. **Simplifying \( \sqrt{1223} \):**
On finding the prime factors of 1223, we note that it is prime itself, hence:
\[
\sqrt{1223} = \sqrt{1223}
\]
**Response:** \( \sqrt{1223} \)

4. **Simplifying \( \sqrt{43 + 27} \):**
\[
\sqrt{43 + 27} = \sqrt{70}
\]
**Response:** \( \sqrt{70} \)

5. **Simplifying \( \sqrt{312 - 23} \):**
\[
\sqrt{312 - 23} = \sqrt{289} = 17
\]
**Response:** \( 17 \)

Now, let’s summarize the simplified results:

1. \( \sqrt{108} = 6\sqrt{3} \)
2. \( \sqrt{15} \cdot 5 = 5\sqrt{15} \)
3. \( \sqrt{1223} = \sqrt{1223} \)
4. \( \sqrt{43 + 27} = \sqrt{70} \)
5. \( \sqrt{312 - 23} = 17 \)

These are the final simplest forms for the given expressions.