-5126 = -6 - 5(x+ 22)^(5/3)

First we will add 6 to both sides:

==> -5120 = -5(x+22)^(5/3)

Now divide by -5:

==> 1024 = (x+ 22)^(5/3)

Now we know that 1024 = 2^10

==> 2^10 = (x+22)^5/3

Now we will raise both sides to the power (3/5)

==> (2^10)^3/5) = [(x+22)^5/3)]^3/5

==> We know that:

a^b^c = a^(bc)

==> 2^(10*3/5) = (x+ 22) ^1

==> 2^(6) = x+ 22

==> 64 = x+ 22

Now subtract 22 from both sides:

==> x= 64 - 22 = 42

**==> x= 42**

We'll move -6 to the left side, changing it's sign;

-5126+6 = - 5(x+ 22)^(5/3)

-5120 = 5(x+ 22)^(5/3)

Since 5120 is divisible by 5, we'll divide by -5 both sides:

1024 = (x+ 22)^(5/3)

We'll raise both sides to 3/5 power:

1024^(3/5) = x+22

We'll use the symmetric property and we'll subtract 22 both sides:

x = 1024^(3/5) - 22

x = 2^(10*3/5) - 22

x = 2^6 - 22

x = 64 - 22

**x = 42**

Since the order of the radical is odd, we don't have constraints for the value of the solution of the equation, so the value for x is accepted.