$textbf{(a)}$      The number 0.000054 has $textbf{two significant figures.}$ The four zeros preceding the digits 54 serve to fix the decimal point.

$textbf{(b)}$      The number $3.001 cdot 10^{5}$ has $textbf{four significant figures.}$ The number of digits that preced the power of 10 determine the number of significant figures.

$textbf{(a)}$      Two significant figures.

$textbf{(b)}$      Four significant figures.

The square root of 2 to four significant figures is

$$
begin{align*}
boxed{sqrt{2}=1.414}\
end{align*}
$$

The digits to the right of the decimal point are all significant, as is the digit to the left of decimal point. Furthermore, nonzero numbers are always significant, and here we have four of them.

$$
begin{align*}
boxed{sqrt{2}=1.414}\
end{align*}
$$