Write each power as the product of the same factor. then find the value

7^2 

Our Factor is: 7
Our Power is: 2

The power determines the number of times it will multiply by that factor.
So in this case: 
7^2 = 7 * 7 = 49

2^8 

Our Factor is:  2
Our Power is: 8

So we need to multiply the factor by itself 8 times. (Think of it as 2 needs to show up 8 times... so if you were to count the number of 2's you'd see there are 8!) 

2*2*2*2*2*2*2*2 = (2*2)(2*2)(2*2)(2*2) = 4*4*4*4 = (4*4)(4*4) = 16*16 = (16*16) = 256

Or if you want to think of it differently you can do the problem like this: 
2^8

2*2 = 4
4*2 = 8
8*2 = 16
16*2 = 32
32*2 = 64
64*2 = 128
128*2 = 256

You should see 7, *2's. (It will always be one less than the power)  

5^5

Our Factor is: 5
Our Power is: 5

5^5 = 5*5*5*5*5 = (5*5)(5*5)*5 = 25*25*5 = (25*5)*25 = 125*25 = 3125

Or you could think of it like this: 
5*5 = 25
25*5 = 125
125*5 = 625
625*5 = 3125