The function f(x) = ax^r satisfies f(2) = 1 and f(32) = 4. Find r.

im just going to accept that i am dumb

What if the solution was a fraction?

f(x)  =  ax^r

f(2)  =  1

a(2)^r  =  1                  Divide both sides by  2^r .

a  =  1 / ( 2^r)

f(32)  =  4

a(32)^r  =  4              Divide both sides by  32^r .

a  =  4 / ( 32^r)

Set these two values of  a  equal to each other.

1 / ( 2^r)  =  4 / (32^r)         Cross multiply.

32^r  =  4 * 2^r                   And we can write  32  and  4  as powers of  2 .

(2^5)^r  =  2^2 * 2^r

2^(5r)  =  2^(2 + r)

5r  =  2 + r

4r  =  2

r  =  1/2