Answer
(a) Please see graphs and explanations below.
(b) $1.2$ and $22.4$
Work Step by Step
(a) First, we compare the rates of growth by graphing the functions $f(x)=2^x$
and $g(x)=x^5$ on the same set of axes as shown in the figures.
(i) This small view window shows the first intersection of the two curves at $(1.2,2.3)$ which gives
the first solution of the equation in (b). In this window, the rate of change for $g(x)$ is higher after
the first intersection point.
(ii) in this enlarged view window, the second intersection at $(22.4,5.7\times10^6)$ can be found.
The rate of change for $f(x)$ appeared to be higher after $x=20$.
(iii) in this wide overview window for the two functions, $f(x)$ clearly wins the race of rate of change
indicated by its much faster growth rate in the vertical direction.
(b) The solutions of the equation $2^x = x^5$ have been found from above (i) and (ii) to be $1.2$ and $22.4$ (rounded to one decimal place).
