Question

How to solve `4^(x^2)=(1/32)^((x+6)/5)` ?

Answer 1

There are no Real solutions to the given equation

Explanation:

`4 = 2^2` and `1/32 = 2^(-5)`

So
`color(white)("XXXX")` `4^(x^2) = (1/32)^((x+6)/5)`
is equivalent to
`color(white)("XXXX")` `(2^2)^(x^2) = (2^(-5))^((x+6)/5)`
or
`color(white)("XXXX")` `2^(2x^2) = 2^(-(x+6))`

So
`color(white)("XXXX")` `2x^2 = -(x+6)`

`color(white)("XXXX")` `2x^2+x+6 = 0`

Since the discriminant (`b^2-4ac`) is less than `0` there are no Real solutions for the given equation.

Graph of `4^(x^2) - (1/32)^((x+6)/5)` Note that it never reaches the x-axis (i.e. it is never zero).
graph{4^(x^2)-(1/32)^((x+6)/5) [-10, 10, -5, 5]}