This is indices rather than surds.

For the first question (I will use "times" in place of x to not confuse the multiplication symbol with the variable x), we change 4 and 8 to base 2.


2^x times 4^(x - 1) = 8^(2x - 1)
2^x times (2^2)^(x - 1) = (2^3)^(2x - 1)


Using the power rule (a^b)^c = a^(bc),

2^x times 2^(2x - 2) = 2^(6x - 3)


Using the same base rule a^b times a^c = a^(b + c),

2^(3x - 2) = 2^(6x - 3)


Now we simply compare the powers.

3x - 2 = 6x - 3
3 - 2 = 6x - 3x
1 = 3x
1/3 = x

For the second question, we do likewise. Change 4 and 8 into base 2.

(4^x)^x = 8^(5x - 6)
[(2^2)^x]^x = (2^3)^(5x - 6)

Using the power rule (a^b)^c = a^(bc),
2^(2x²) = 2^(15x - 18)

Now we simply compare the powers.

2x² = 15x - 18
2x² - 15x + 18 = 0
(2x - 3) (x - 6) = 0
2x - 3 = 0 or x - 6 = 0
x = 3/2 or x = 6

Both answers are equally valid.

For the third one, note that 2.5 = 5/2 and 0.4 = 2/5.

These are inverses of each other, so we can write 0.4 as (2.5)^-1.

We proceed as usual.


2.5^(x - 3) = 0.4^x
2.5^(x - 3) = [2.5^(-1)]^x


Using the power rule (a^b)^c = a^(bc),
2.5^(x - 3) = 2.5^(-x)


We simply compare the powers.
x - 3 = -x
2x = 3
x = 3/2

Thank you sorry it was labeled as surds