Skeeter's method, dividing both sides by 9/2 first, is simpler than yours but it can be done your way:
I don't know how you now got ".125x-4.5=x". Was that first "x" supposed to be a multiplication sign? Bad idea! Never use the same symbol to mean two different things!
But even so, you get rid of the "9/2" power on the left by taking the 9/2 root- which is the same thing as the 2/9 power, on both sides:
If you want or are required to "rationalize the denominator", multiply both numerator and denominator by : .