Perform the indicated operation, leaving all answers in simplest form.(5 points)

Question

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

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

Answer 1

Here are the simplifications for each operation:

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

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

3. \( \sqrt{1223} \): Since 1223 cannot be simplified further as it does not have square factors, we keep it as is. Response: \( \sqrt{1223} \)

4. \( \sqrt{43+27} \): \[ 43 + 27 = 70 \quad \text{and} \quad \sqrt{70} \text{ cannot be simplified further.} \] Response: \( \sqrt{70} \)

5. \( \sqrt{312} - \sqrt{23} \): Simplifying \( \sqrt{312} = \sqrt{4 \cdot 78} = 2\sqrt{78} \): \[ 2\sqrt{78} - \sqrt{23} \text{ cannot be simplified further.} \] Response: \( 2\sqrt{78} - \sqrt{23} \)

6. \( \sqrt{23} - \sqrt{43} \): These terms do not simplify; they remain as they are. Response: \( \sqrt{23} - \sqrt{43} \)

7. \( \sqrt{312} \) again: This was already calculated. Response: \( 2\sqrt{78} \)

8. \( 6\sqrt{3} \) again: This was already calculated. Response: \( 6\sqrt{3} \)

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

10. \( 2\sqrt{3} - \sqrt{312 - 23} \): First, simplify \( 312 - 23 = 289 \) so: \[ \sqrt{289} = 17 \] Therefore: \[ 2\sqrt{3} - 17 \] Response: \( 2\sqrt{3} - 17 \)

11. \( \sqrt{73} - \sqrt{43} \): These terms cannot be simplified further; they remain as they are. Response: \( \sqrt{73} - \sqrt{43} \)

12. \( \sqrt{43} + 27 \): This cannot have anything simplified, so it remains. Response: \( \sqrt{43} + 27 \)

13. \( \sqrt{53} \): This is already in simplest form as well. Response: \( \sqrt{53} \)

14. \( \sqrt{63} \): \[ \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \] Response: \( 3\sqrt{7} \)

15. \( \sqrt{33} \): This is already in simplest form too. Response: \( \sqrt{33} \)

16. \( \sqrt{83} \): This one too cannot be simplified. Response: \( \sqrt{83} \)

You can plug in the responses accordingly based on this fundamental arithmetic and simplification.