To evaluate the given equation `4^(2x-5)=64^(3x)` , we may let `64 =4^3` .

The equation becomes: `4^(2x-5)=(4^3)^(3x)` .

Apply Law of exponents: `(x^n)^m = x^(n*m)` .

`4^(2x-5)=4^(3*3x)`

`4^(2x-5)=4^(9x)`

Apply the theorem: If `b^x=b^y` then `x=y` .

If `4^(2x-5)=4^(9x` ) then `2x-5=9x` .

Subtract 2x on both sides of the equation `2x-5=9x` .

`2x-5-2x=9x-2x`

`-5=7x`

Divide both sides by `7` .

`(-5)/7=(7x)/7`

`x = -5/7`

Checking: Plug-in `x=-5/7` on `4^(2x-5)=64^(3x).`

`4^(2(-5/7)-5)=?64^(3*(-5/7))`

`4^((-10)/7-5)=?64^((-15)/7)`

`4^((-45)/7)=?64^((-15)/7)`

`4^((-45)/7)=?(4^3)^((-15)/7)`

`4^((-45)/7)=?4^(3*(-15)/7)`

`4^((-45)/7)=4^((-45)/7)` **TRUE**

or

`0.000135~~0.000135` **TRUE**

Thus, the `x=-5/7` is the **real exact solution** of the
equation `4^(2x-5)=64^(3x)` .

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