Fractional indeces help maths question?

2 √8 = 2^m

First, I'll simplify the radical. I can factor out a 4 from the 8 and pull it out so we get:

2 √(4 * 2) = 2^m
2 * 2 √2 = 2^m

Now we can convert the radical into a fractional exponent of 1/2:

2 * 2 * 2^(1/2) = 2^m

Now if we multiply numbers of the same base together, it's the same as adding the exponents. We have two 2's with an exponent of 1, and one with 1/2, add them together we have:

2^(5/2) = 2^m

Now if we have two numbers equal to each other with the same bases, the exponents must also be the same.

m = 5/2

2^m = 2 x 2 x √2

2^m = 2^(5/2)

m = 5/2

2^m = 2*sqrt(8)
m = log (base 2) of (2*sqrt(8))
m =~ 2.5