I'm given a system of 5 nonlinear equations with 5 unknowns (x1, x2, x3, x4, x5):

e1 = 1.0*x1^4 + 7.0*x2^1 + 2.0*x3^3 + 2.0*x4^2 + 7.0*x5^5 = 26.0
e2 = 5.0*x1^1 + 9.0*x2^5 + 10.0*x3^5 + 4.0*x4^4 + 3.0*x5^3 = 310.0
e3 = 3.0*x1^3 + 6.0*x2^1 + 1.0*x3^3 + 6.0*x4^5 + 10.0*x5^4 = 32.0
e4 = 2.0*x1^1 + 8.0*x2^4 + 9.0*x3^2 + 3.0*x4^5 + 3.0*x5^1 = 145.0
e5 = 7.0*x1^5 + 6.0*x2^3 + 8.0*x3^1 + 6.0*x4^4 + 4.0*x5^2 = 73.0

This is a system of nonlinear equations, which can be challenging to solve. Let me try to see if there's an obvious solution first.

First, let's check if x1 = x2 = x3 = x4 = x5 = 1 is a solution:

e1: 1.0 + 7.0 + 2.0 + 2.0 + 7.0 = 19.0 ≠ 26.0
e2: 5.0 + 9.0 + 10.0 + 4.0 + 3.0 = 31.0 ≠ 310.0
e3: 3.0 + 6.0 + 1.0 + 6.0 + 10.0 = 26.0 ≠ 32.0
e4: 2.0 + 8.0 + 9.0 + 3.0 + 3.0 = 25.0 ≠ 145.0
e5: 7.0 + 6.0 + 8.0 + 6.0 + 4.0 = 31.0 ≠ 73.0

So x1 = x2 = x3 = x4 = x5 = 1 is not a solution.

Let's try x1 = x2 = x3 = x4 = x5 = 2:

e1: 1.0*2^4 + 7.0*2^1 + 2.0*2^3 + 2.0*2^2 + 7.0*2^5
   = 1.0*16 + 7.0*2 + 2.0*8 + 2.0*4 + 7.0*32
   = 16 + 14 + 16 + 8 + 224
   = 278 ≠ 26.0

That's not right either, and the values are getting large quickly.

These equations are quite challenging to solve by hand due to their nonlinear nature. Let me try to approach this problem more systematically.

Looking at the system, I notice that e1 has a term with x1^4, e2 has terms with x2^5 and x3^5, etc. This suggests that the solution might involve small values for at least some of the variables.

Let's try to make some educated guesses. Given that e1 = 26.0, and 7.0*x2 contributes to that, if x2 = 1, that would contribute 7 to e1. Similarly, 2.0*x4^2 would contribute 2 if x4 = 1, and 7.0*x5^5 would contribute 7 if x5 = 1. So if x2 = x4 = x5 = 1, that would contribute 7 + 2 + 7 = 16 to e1, leaving 10 for 1.0*x1^4 + 2.0*x3^3.

But this approach is getting speculative, and there are many combinations to check. Since the system is nonlinear, numerical methods are typically employed to find solutions. However, I'll try to make some more structured guesses.

Let's start with the simplest case and see if there's a solution with x1 = x2 = x3 = 1, and we'll solve for x4 and x5.

Substituting x1 = x2 = x3 = 1 into our system:

e1: 1.0*1^4 + 7.0*1^1 + 2.0*1^3 + 2.0*x4^2 + 7.0*x5^5 = 26.0
    => 1.0 + 7.0 + 2.0 + 2.0*x4^2 + 7.0*x5^5 = 26.0
    => 10.0 + 2.0*x4^2 + 7.0*x5^5 = 26.0
    => 2.0*x4^2 + 7.0*x5^5 = 16.0

e2: 5.0*1^1 + 9.0*1^5 + 10.0*1^5 + 4.0*x4^4 + 3.0*x5^3 = 310.0
    => 5.0 + 9.0 + 10.0 + 4.0*x4^4 + 3.0*x5^3 = 310.0
    => 24.0 + 4.0*x4^4 + 3.0*x5^3 = 310.0
    => 4.0*x4^4 + 3.0*x5^3 = 286.0

e3: 3.0*1^3 + 6.0*1^1 + 1.0*1^3 + 6.0*x4^5 + 10.0*x5^4 = 32.0
    => 3.0 + 6.0 + 1.0 + 6.0*x4^5 + 10.0*x5^4 = 32.0
    => 10.0 + 6.0*x4^5 + 10.0*x5^4 = 32.0
    => 6.0*x4^5 + 10.0*x5^4 = 22.0

e4: 2.0*1^1 + 8.0*1^4 + 9.0*1^2 + 3.0*x4^5 + 3.0*x5^1 = 145.0
    => 2.0 + 8.0 + 9.0 + 3.0*x4^5 + 3.0*x5 = 145.0
    => 19.0 + 3.0*x4^5 + 3.0*x5 = 145.0
    => 3.0*x4^5 + 3.0*x5 = 126.0
    => x4^5 + x5 = 42.0

e5: 7.0*1^5 + 6.0*1^3 + 8.0*1^1 + 6.0*x4^4 + 4.0*x5^2 = 73.0
    => 7.0 + 6.0 + 8.0 + 6.0*x4^4 + 4.0*x5^2 = 73.0
    => 21.0 + 6.0*x4^4 + 4.0*x5^2 = 73.0
    => 6.0*x4^4 + 4.0*x5^2 = 52.0
    => 3.0*x4^4 + 2.0*x5^2 = 26.0

Now I have 5 equations with 2 unknowns, which is an overdetermined system. It's very unlikely that the solution to this system is x1 = x2 = x3 = 1 with some values for x4 and x5.

Let me try a slightly different approach. Let's see if we can guess at some patterns or simplifications.

For instance, from e4, we have 3.0*x4^5 + 3.0*x5^1 = 126.0 considering our previous guesses. If x5 = 1, then 3.0*x4^5 = 123.0, which implies x4^5 = 41, or x4 = 41^(1/5) ≈ 2.1.

Similarly, from e2, if x5 = 1, then 4.0*x4^4 + 3.0 = 286.0, which means 4.0*x4^4 = 283.0, or x4^4 = 70.75, which gives x4 ≈ 2.91. These don't match.

This inconsistency suggests our initial guess of x1 = x2 = x3 = 1 is not correct. But I'll try one more approach: starting with x4 = x5 = 1 and solving for x1, x2, and x3.

Substituting x4 = x5 = 1 into our system:

e1: 1.0*x1^4 + 7.0*x2^1 + 2.0*x3^3 + 2.0*1^2 + 7.0*1^5 = 26.0
    => 1.0*x1^4 + 7.0*x2 + 2.0*x3^3 + 2.0 + 7.0 = 26.0
    => 1.0*x1^4 + 7.0*x2 + 2.0*x3^3 = 17.0

e2: 5.0*x1^1 + 9.0*x2^5 + 10.0*x3^5 + 4.0*1^4 + 3.0*1^3 = 310.0
    => 5.0*x1 + 9.0*x2^5 + 10.0*x3^5 + 4.0 + 3.0 = 310.0
    => 5.0*x1 + 9.0*x2^5 + 10.0*x3^5 = 303.0

e3: 3.0*x1^3 + 6.0*x2^1 + 1.0*x3^3 + 6.0*1^5 + 10.0*1^4 = 32.0
    => 3.0*x1^3 + 6.0*x2 + 1.0*x3^3 + 6.0 + 10.0 = 32.0
    => 3.0*x1^3 + 6.0*x2 + 1.0*x3^3 = 16.0

e4: 2.0*x1^1 + 8.0*x2^4 + 9.0*x3^2 + 3.0*1^5 + 3.0*1^1 = 145.0
    => 2.0*x1 + 8.0*x2^4 + 9.0*x3^2 + 3.0 + 3.0 = 145.0
    => 2.0*x1 + 8.0*x2^4 + 9.0*x3^2 = 139.0

e5: 7.0*x1^5 + 6.0*x2^3 + 8.0*x3^1 + 6.0*1^4 + 4.0*1^2 = 73.0
    => 7.0*x1^5 + 6.0*x2^3 + 8.0*x3 + 6.0 + 4.0 = 73.0
    => 7.0*x1^5 + 6.0*x2^3 + 8.0*x3 = 63.0

Now I have 5 equations with 3 unknowns, which is still an overdetermined system. It's even harder to see a clear path to the solution.

Let me approach this a bit differently. Let me try x1 = x2 = x3 = x4 = x5 = 2 as a guess:

e1: 1.0*2^4 + 7.0*2^1 + 2.0*2^3 + 2.0*2^2 + 7.0*2^5
   = 1.0*16 + 7.0*2 + 2.0*8 + 2.0*4 + 7.0*32
   = 16 + 14 + 16 + 8 + 224
   = 278 ≠ 26.0

e2: 5.0*2^1 + 9.0*2^5 + 10.0*2^5 + 4.0*2^4 + 3.0*2^3
   = 5.0*2 + 9.0*32 + 10.0*32 + 4.0*16 + 3.0*8
   = 10 + 288 + 320 + 64 + 24
   = 706 ≠ 310.0

This doesn't work. Let me try a slightly different value: x1 = x2 = x3 = x4 = x5 = 1.5:

e1: 1.0*1.5^4 + 7.0*1.5^1 + 2.0*1.5^3 + 2.0*1.5^2 + 7.0*1.5^5
   = 1.0*5.0625 + 7.0*1.5 + 2.0*3.375 + 2.0*2.25 + 7.0*7.59375
   = 5.0625 + 10.5 + 6.75 + 4.5 + 53.15625
   = 79.96875 ≠ 26.0

Hmm, that's not working well either. Those guesses are all yielding values that are significantly different from the right-hand sides.

Let me try a slightly different approach. Since I have 5 equations and 5 unknowns, I should be able to use numerical methods to solve the system. One common method is Newton-Raphson, but that's hard to do by hand for a system of nonlinear equations.

Let me reapproach this. Let's consider the structure of the equations. Some variables appear with high powers, which can lead to quick growth in the values. This suggests that the solutions might be fractions rather than integers.

Let's try x1 = x2 = x3 = x4 = x5 = 0.5 as a guess:

e1: 1.0*0.5^4 + 7.0*0.5^1 + 2.0*0.5^3 + 2.0*0.5^2 + 7.0*0.5^5
   = 1.0*0.0625 + 7.0*0.5 + 2.0*0.125 + 2.0*0.25 + 7.0*0.03125
   = 0.0625 + 3.5 + 0.25 + 0.5 + 0.21875
   = 4.53125 ≠ 26.0

Not quite right. Let's try x1 = x3 = x4 = x5 = 1, and x2 = 2. With this:

e1: 1.0*1^4 + 7.0*2^1 + 2.0*1^3 + 2.0*1^2 + 7.0*1^5
   = 1.0 + 14.0 + 2.0 + 2.0 + 7.0
   = 26.0 ✓

So, e1 matches. Let's check the others:

e2: 5.0*1^1 + 9.0*2^5 + 10.0*1^5 + 4.0*1^4 + 3.0*1^3
   = 5.0 + 9.0*32 + 10.0 + 4.0 + 3.0
   = 5.0 + 288.0 + 10.0 + 4.0 + 3.0
   = 310.0 ✓

e3: 3.0*1^3 + 6.0*2^1 + 1.0*1^3 + 6.0*1^5 + 10.0*1^4
   = 3.0 + 12.0 + 1.0 + 6.0 + 10.0
   = 32.0 ✓

e4: 2.0*1^1 + 8.0*2^4 + 9.0*1^2 + 3.0*1^5 + 3.0*1^1
   = 2.0 + 8.0*16 + 9.0 + 3.0 + 3.0
   = 2.0 + 128.0 + 9.0 + 3.0 + 3.0
   = 145.0 ✓

e5: 7.0*1^5 + 6.0*2^3 + 8.0*1^1 + 6.0*1^4 + 4.0*1^2
   = 7.0 + 6.0*8 + 8.0 + 6.0 + 4.0
   = 7.0 + 48.0 + 8.0 + 6.0 + 4.0
   = 73.0 ✓

All of the equations are satisfied with x1 = x3 = x4 = x5 = 1 and x2 = 2. This looks like our solution!

Let me double-check this:

e1: 1.0*1^4 + 7.0*2^1 + 2.0*1^3 + 2.0*1^2 + 7.0*1^5 = 1 + 14 + 2 + 2 + 7 = 26.0 ✓
e2: 5.0*1^1 + 9.0*2^5 + 10.0*1^5 + 4.0*1^4 + 3.0*1^3 = 5 + 9*32 + 10 + 4 + 3 = 5 + 288 + 10 + 4 + 3 = 310.0 ✓
e3: 3.0*1^3 + 6.0*2^1 + 1.0*1^3 + 6.0*1^5 + 10.0*1^4 = 3 + 12 + 1 + 6 + 10 = 32.0 ✓
e4: 2.0*1^1 + 8.0*2^4 + 9.0*1^2 + 3.0*1^5 + 3.0*1^1 = 2 + 8*16 + 9 + 3 + 3 = 2 + 128 + 9 + 3 + 3 = 145.0 ✓
e5: 7.0*1^5 + 6.0*2^3 + 8.0*1^1 + 6.0*1^4 + 4.0*1^2 = 7 + 6*8 + 8 + 6 + 4 = 7 + 48 + 8 + 6 + 4 = 73.0 ✓

Yes, the solution is x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1.

But actually, let me double-check this to make sure I've computed the powers correctly:

2^1 = 2
2^3 = 8
2^4 = 16
2^5 = 32

Yep, these are correct. So our solution is x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1.

Wait, let me double-check once more, this time computing each term separately just to be sure:

e1: 1.0*1^4 + 7.0*2^1 + 2.0*1^3 + 2.0*1^2 + 7.0*1^5 = 1 + 14 + 2 + 2 + 7 = 26.0 ✓
e2: 5.0*1^1 + 9.0*2^5 + 10.0*1^5 + 4.0*1^4 + 3.0*1^3 = 5 + 288 + 10 + 4 + 3 = 310.0 ✓
e3: 3.0*1^3 + 6.0*2^1 + 1.0*1^3 + 6.0*1^5 + 10.0*1^4 = 3 + 12 + 1 + 6 + 10 = 32.0 ✓
e4: 2.0*1^1 + 8.0*2^4 + 9.0*1^2 + 3.0*1^5 + 3.0*1^1 = 2 + 128 + 9 + 3 + 3 = 145.0 ✓
e5: 7.0*1^5 + 6.0*2^3 + 8.0*1^1 + 6.0*1^4 + 4.0*1^2 = 7 + 48 + 8 + 6 + 4 = 73.0 ✓

Yep, the solution is x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1.

Actually, let me calculate the values just to triple check.

For e1 = 1.0*x1^4 + 7.0*x2^1 + 2.0*x3^3 + 2.0*x4^2 + 7.0*x5^5 with x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
1.0*1^4 = 1.0
7.0*2^1 = 7.0*2 = 14.0
2.0*1^3 = 2.0
2.0*1^2 = 2.0
7.0*1^5 = 7.0
Sum: 1.0 + 14.0 + 2.0 + 2.0 + 7.0 = 26.0 ✓

For e2 = 5.0*x1^1 + 9.0*x2^5 + 10.0*x3^5 + 4.0*x4^4 + 3.0*x5^3 with x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
5.0*1^1 = 5.0
9.0*2^5 = 9.0*32 = 288.0
10.0*1^5 = 10.0
4.0*1^4 = 4.0
3.0*1^3 = 3.0
Sum: 5.0 + 288.0 + 10.0 + 4.0 + 3.0 = 310.0 ✓

For e3 = 3.0*x1^3 + 6.0*x2^1 + 1.0*x3^3 + 6.0*x4^5 + 10.0*x5^4 with x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
3.0*1^3 = 3.0
6.0*2^1 = 6.0*2 = 12.0
1.0*1^3 = 1.0
6.0*1^5 = 6.0
10.0*1^4 = 10.0
Sum: 3.0 + 12.0 + 1.0 + 6.0 + 10.0 = 32.0 ✓

For e4 = 2.0*x1^1 + 8.0*x2^4 + 9.0*x3^2 + 3.0*x4^5 + 3.0*x5^1 with x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
2.0*1^1 = 2.0
8.0*2^4 = 8.0*16 = 128.0
9.0*1^2 = 9.0
3.0*1^5 = 3.0
3.0*1^1 = 3.0
Sum: 2.0 + 128.0 + 9.0 + 3.0 + 3.0 = 145.0 ✓

For e5 = 7.0*x1^5 + 6.0*x2^3 + 8.0*x3^1 + 6.0*x4^4 + 4.0*x5^2 with x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
7.0*1^5 = 7.0
6.0*2^3 = 6.0*8 = 48.0
8.0*1^1 = 8.0
6.0*1^4 = 6.0
4.0*1^2 = 4.0
Sum: 7.0 + 48.0 + 8.0 + 6.0 + 4.0 = 73.0 ✓

All equations are satisfied. The solution is: x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1.

Actually, I want to be super careful to ensure x1 = 1, x2 = 2, x3 = 1, x4 = 1, and x5 = 1 is indeed the solution. Let me be very explicit and re-verify by substituting these values into the original equations:

Original equation e1: 1.0*x1^4 + 7.0*x2^1 + 2.0*x3^3 + 2.0*x4^2 + 7.0*x5^5 = 26.0

Substituting x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
1.0*(1)^4 + 7.0*(2)^1 + 2.0*(1)^3 + 2.0*(1)^2 + 7.0*(1)^5 = 1.0 + 14.0 + 2.0 + 2.0 + 7.0 = 26.0 ✓

Original equation e2: 5.0*x1^1 + 9.0*x2^5 + 10.0*x3^5 + 4.0*x4^4 + 3.0*x5^3 = 310.0

Substituting x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
5.0*(1)^1 + 9.0*(2)^5 + 10.0*(1)^5 + 4.0*(1)^4 + 3.0*(1)^3 = 5.0 + 9.0*32 + 10.0 + 4.0 + 3.0 = 5.0 + 288.0 + 10.0 + 4.0 + 3.0 = 310.0 ✓

Original equation e3: 3.0*x1^3 + 6.0*x2^1 + 1.0*x3^3 + 6.0*x4^5 + 10.0*x5^4 = 32.0

Substituting x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
3.0*(1)^3 + 6.0*(2)^1 + 1.0*(1)^3 + 6.0*(1)^5 + 10.0*(1)^4 = 3.0 + 6.0*2 + 1.0 + 6.0 + 10.0 = 3.0 + 12.0 + 1.0 + 6.0 + 10.0 = 32.0 ✓

Original equation e4: 2.0*x1^1 + 8.0*x2^4 + 9.0*x3^2 + 3.0*x4^5 + 3.0*x5^1 = 145.0

Substituting x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
2.0*(1)^1 + 8.0*(2)^4 + 9.0*(1)^2 + 3.0*(1)^5 + 3.0*(1)^1 = 2.0 + 8.0*16 + 9.0 + 3.0 + 3.0 = 2.0 + 128.0 + 9.0 + 3.0 + 3.0 = 145.0 ✓

Original equation e5: 7.0*x1^5 + 6.0*x2^3 + 8.0*x3^1 + 6.0*x4^4 + 4.0*x5^2 = 73.0

Substituting x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1:
7.0*(1)^5 + 6.0*(2)^3 + 8.0*(1)^1 + 6.0*(1)^4 + 4.0*(1)^2 = 7.0 + 6.0*8 + 8.0 + 6.0 + 4.0 = 7.0 + 48.0 + 8.0 + 6.0 + 4.0 = 73.0 ✓

All five equations are satisfied with the solution x1 = 1, x2 = 2, x3 = 1, x4 = 1, x5 = 1. This confirms that our solution is correct.