Just to chime in, in case you haven't studied calculus yet, there's always the bisection method. Let
f(x) = x^2 - 4^x
You want to find where f(x) = 0. Since
f(-1) > 0
f(0) < 0
f(x) = 0 for some x in the interval [-1,0]
so, start guessing at the center of the interval. pick whichever half of the interval the change of sign appears in.
f(-0.75) = 0.2089
f(-0.625) = -0.2982
f(-0.6875) = 0.0871
f(-0.65625) = 0.2804
... 9 steps later ...
f(-0.6412353515625) = 0.000091888
As you can see, it converges rather more slowly than Newton's method. It took 13 iterations to get the same accuracy of 3 steps with Newton's method.