Although that looks hard, it can be done without a calculator by
partitioning $6$ into $2.5+1.9+1.6$, and finding appropriate powers
of each number. The last one is easy for computer geeks like me who
have memorized many small powers of $2$.
This, too, is pretty easy without a calculator - $4.096^{1/3}$ is
$1.6$, and the square root of $2$ is more than $1.4$, so the first
sum is more than $5$, and clearly less than the second sum.
Thanks for the extensions, Graeme.
These inequalities show that the graph is
going to intersect the x-axis somewhere between 4 and 7, which it
does: