Solve for X

Mutiply both sides by the denominator on the left

l 5 + ^x√-177147 l = (2√5 - 4) (√5 + 2)

l 5 + ^x√-177147 l = 10 - 8

l 5 + ^x√-177147 l = 2   

This says that either

-(5 + ^x√-177147 )  = 2     or  (5 + ^x√-177147 )  = 2

Working with the first we have

- ^x√-177147  = 7

^x√-177147  = -7

-177177 = (-7)^x   .....and this has no real solutions

Working with the second, we have

 ^x√-177147 = -3

(-3)^x = -177147

Note that, we cannot take the log of both sides here because log(-3)^x would be undefined as well as log(-177147)

However.......if this has an integer solution, x must be an odd number....

So we can write

(-1)^x (3)^x  = -177147

And, if x is an odd integer, (-1)^x = -1.....so we have

(-1) * (3)^x  = -177147   divide both sides by -1

(3)^x  = 177147

Now...take the log of both sides....and using a log property, we have

x log3  = log 177147   divide both sides by log 3

x = log 177147 / log 3 =  11

first multiply both sides by the denominator on the right.

Then subtract 5 from both sides.

This should give you a radical to the x^th root of -177147 = -3

Then take the log of both sides. ..and multiply by -1.

log 177147=log 3^x

5.248=x log 3

x= 11