1. **State the problem:** Calculate the square of the number $1.5 \times 10^{-2}$. 2. **Formula used:** To square a number in scientific notation, use $\left(a \times 10^{b}\right)^2 = a^2 \times 10^{2b}$. 3. **Apply the formula:** $$\left(1.5 \times 10^{-2}\right)^2 = 1.5^2 \times 10^{2 \times (-2)}$$ 4. **Calculate the powers:** $$1.5^2 = 2.25$$ $$10^{2 \times (-2)} = 10^{-4}$$ 5. **Combine the results:** $$2.25 \times 10^{-4}$$ 6. **Final answer:** $$\boxed{2.25 \times 10^{-4}}$$