Step 1: Understand the surface involved.
A soap bubble is a thin liquid film with air inside and air outside, so it has two free surfaces. Any work done in blowing it up goes into creating extra surface area against surface tension $T$.
Step 2: Write the work relation.
Work done equals surface tension times the total increase in area: $$W = T \,\Delta A_{\text{total}}.$$ Because there are two surfaces, $\Delta A_{\text{total}} = 2 \times 4\pi\left(r_2^2 - r_1^2\right) = 8\pi\left(r_2^2 - r_1^2\right).$
Step 3: Convert the data to SI units.
$T = 0.03\ \text{Nm}^{-1}$, $r_1 = 3\ \text{cm} = 0.03\ \text{m}$, and $r_2 = 5\ \text{cm} = 0.05\ \text{m}$.
Step 4: Evaluate the difference of squares.
$$r_2^2 - r_1^2 = (0.05)^2 - (0.03)^2 = 0.0025 - 0.0009 = 0.0016\ \text{m}^2.$$
Step 5: Plug everything into the formula.
$$W = 8\pi \times 0.03 \times 0.0016 = 0.24\pi \times 0.0016 = 3.84 \times 10^{-4}\,\pi\ \text{J}.$$
Step 6: Convert joules to millijoules.
Multiplying by $1000$ gives $W = 0.384\pi\ \text{mJ} \approx 0.4\pi\ \text{mJ}.$
\[ \boxed{W \approx 0.4\pi\ \text{mJ}} \]