Calculation - the hard way

Question

This is a puzzle I love to play with my math students and I hope you will enjoy it too:

You are given the numbers 1, 2, 3, 4, and 5 exactly once.

Your target is a number, e.g. 36. Can you create a calculation with addition, subtraction, multiplication, division and parentheses, so that you arrive at this number?

Concatenating the numbers (like 12 out of 1 and 2) is explicitly forbidden.

For this example, the solution would be

$$ 36=(2+4) \cdot (1+5) $$

but remember: You can use each number only once.

Can you find a calculation for all numbers from 1 to 75?

BONUS question: Now powers are allowed! Can you go to 125 now?

Answer 1

1 = 1
2 = 2
3 = 3
4 = 4
5 = 5
6 = 5 + 1
7 = 5 + 2
8 = 5 + 3
9 = 5 + 4
10 = 5 + 4 + 1
11 = 5 + 4 + 2
12 = 5 + 4 + 3
13 = 5 + 4 + 3 + 1
14 = 5 + 4 + 3 + 2
15 = 5 + 4 + 3 + 2 + 1
16 = 4 ⋅ (1 + 3)
17 = (5 ⋅ 4) - 3
18 = (5 ⋅ 4) - 2
19 = (5 ⋅ 4) - 1
20 = (5 ⋅ 4)
21 = (5 ⋅ 4) + 1
22 = (5 ⋅ 4) + 2
23 = (5 ⋅ 4) + 3
24 = (5 + 1) ⋅ 4
25 = ((4 + 5) ⋅ 3) - 2
26 = ((4 + 5) ⋅ 3) - 1
27 = ((4 + 5) ⋅ 3)
28 = ((4 + 5) ⋅ 3) + 1
29 = ((4 + 5) ⋅ 3) + 2
30 = 2 ⋅ 3 ⋅ 5
31 = (2 ⋅ 3 ⋅ 5) + 1
32 = (1 + 3) ⋅ 4 ⋅ 2
33 = (2 ⋅ 3 ⋅ 5) + 4 - 1
34 = (2 ⋅ 3 ⋅ 5) + 4
35 = (2 ⋅ 3 ⋅ 5) + 4 + 1
36 (given above)
37 = ((4 + 3) ⋅ 5) + 2
38 = ((4 + 3) ⋅ 5) + 2 + 1
40 = 2 ⋅ 4 ⋅ 5
41 = (2 ⋅ 4 ⋅ 5) + 1
42 = (2 ⋅ 4 ⋅ 5) - 1 + 3
43 = (2 ⋅ 4 ⋅ 5) + 3
44 = (2 ⋅ 4 ⋅ 5) + 3 + 1
45 = (2 + 3 + 4) ⋅ 5
46 = ((2 + 3 + 4) ⋅ 5) + 1
47 = ((2 + 4) ⋅ (3 + 5)) - 1
48 = ((2 + 4) ⋅ (3 + 5))
49 = ((2 + 4) ⋅ (3 + 5)) + 1
50 = 2 ⋅ 5 ⋅ (4 + 1)
51 = ((5 ⋅ 4) - 2 - 1) ⋅ 3
52 = ((5 ⋅ 2) + 3) ⋅ 4
53 = (((5 ⋅ 2) + 3) ⋅ 4) + 1
54 = (5 + 4) ⋅ 3 ⋅ 2
55 = ((5 + 4) ⋅ 3 ⋅ 2) + 1
56 = (4 + 2 + 1) ⋅ (5 + 3)
57 = (5 ⋅ 4 ⋅ 3) - 2 - 1
58 = (5 ⋅ 4 ⋅ 3) - 2
59 = (5 ⋅ 4 ⋅ 3) - 1
60 = (5 ⋅ 4 ⋅ 3)
61 = (5 ⋅ 4 ⋅ 3) + 1
62 = (5 ⋅ 4 ⋅ 3) + 2
63 = (5 ⋅ 4 ⋅ 3) + 1 + 2
64 = (5 + 3) ⋅ 4 ⋅ 2
65 = ((5 + 3) ⋅ 4 ⋅ 2) + 1
66 = (5 + 1) ⋅ ((4 ⋅ 2) + 3)
67 = ((5 ⋅ 3) + 2) ⋅ 4) - 1
68 = ((5 ⋅ 3) + 2) ⋅ 4)
69 = ((5 ⋅ 3) + 2) ⋅ 4) + 1
70 = (3 + 4) ⋅ 5 ⋅ 2
71 = ((3 + 4) ⋅ 5 ⋅ 2) + 1
72 = (5 + 1) ⋅ 3 ⋅ 4
73 = ((4 + 1) ⋅ 3 ⋅ 5) - 2
74 = ((5 + 1) ⋅ 3 ⋅ 4) + 2
75 = (4 + 1) ⋅ 3 ⋅ 5

These are all the solutions for the main puzzle, I'm going to give the bonus puzzle a go when I wake up.

Answer 2

"a guy" has covered the initial question here is the bonus

$76 = 3^4 - 5$
$77 = 3^4 + 1 - 5$
$78 = 3^4 + 2 - 5$
$79 = 3^4 - 2$
$80 = 3^4 - 1$
$81 = 3^4$
$82 = 3^4 + 1$
$83 = 3^4 + 2$
$84 = 3^4 + 2 + 1$
$85 = 3^4 + 5 - 1$
$86 = 3^4 + 5$
$87 = 3^4 + 5 + 1$
$88 = 3^4 + 5 + 2$
$89 = 3^4 + 5 + 2 + 1$
$90 = 3^4 + (5\times 2) - 1$
$91 = 3^4 + (5\times 2)$
$92 = 3^4 + (5\times 2) + 1$
$93 = 3^4 + ((5+1)\times 2)$
$94 = ((2^4 + 3) \times 5) - 1$
$95 = ((2^4 + 3) \times 5)$
$96 = ((2^4 + 3) \times 5) + 1$
$97 = (2^5 \times 3) + 1$
$98 = (5^2 \times 4) - 3 + 1$
$99 = (5^2 \times 4) - 1$
$100 = (5^2 \times 4)$
$101 = (5^2 \times 4) + 1$
$102 = (5^2 \times 4) + 3 - 1$
$103 = (5^2 \times 4) + 3$
$104 = (5^2 \times 4) + 3 + 1$
$105 = 3 \times 5 \times (1+2+4)$
$106 = 3^4 + 5^2$
$107 = 3^4 + 5^2 + 1$
$108 = (5^2 + 3 - 1) \times 4$
$109 = ((2^5 + 4) \times 3) + 1$
$110 = ((4\times 3) - 1) \times 2 \times 5$
$111 = ((5^2 + 3) \times 4) - 1$
$112 = ((5^2 + 3) \times 4)$
$113 = ((5^2 + 3) \times 4) + 1$
$114 = ((5\times 4)-1) \times 3 \times 2$
$115 = 5^3 - ((4+1)\times 2)$
$116 = 5^3 - (4\times 2) - 1$
$117 = 5^3 - (4\times 2)$
$118 = 5^3 - (4\times 2) + 1$
$119 = 5^3 - 4 - 2$
$120 = 5^3 - 4 - 1$
$121 = 5^3 - 4$
$122 = 5^3 - 2 - 1$
$123 = 5^3 - 2$
$124 = 5^3 - 1$
$125 = 5^3$

We can even go a bit further, without much difficulty

$126 = 5^3 + 1$
$127 = 5^3 + 2$
$128 = 5^3 + 2 + 1$
$129 = 5^3 + 4$
$130 = 5^3 + 4 + 1$
$131 = 5^3 + 4 + 2$
$132 = 5^3 + 4 + 2 + 1$
$133 = 5^3 + (4\times 2)$
$134 = 5^3 + (4\times 2) + 1$
$135 = 5^3 + ((4+1) \times 2)$

Here are a few more

$136 = (2^5 + 3 - 1) \times 4$
$137 = 5^3 + (4 \times (2+1))$
$138 = (4^3 + 5) \times 2$
$139 = ((4^3 + 5) \times 2) + 1$
$140 = (4^3 + 5 + 1) \times 2$
$141 = 5^3 + 2^4$
$142 = 5^3 + 2^4 + 1$
$143 = (3 \times 4)^2 - 1$
$144 = (3 \times 4)^2$
$145 = (3 \times 4)^2 + 1$