1. The problem is to estimate the square root of 0.036. 2. Recall the formula for square root: if $x = a^2$, then $\sqrt{x} = a$. 3. Notice that 0.036 can be written as $\frac{36}{1000}$ or $\frac{36}{100}$ since $0.036 = 36 \times 10^{-3}$. 4. More simply, $0.036 = 0.06^2$ because $0.06 \times 0.06 = 0.0036$ (which is incorrect, so let's check carefully). 5. Let's try $0.06 \times 0.06 = 0.0036$, so that's too small. 6. Try $0.19 \times 0.19 = 0.0361$, close to 0.036. 7. Try $0.18 \times 0.18 = 0.0324$, less than 0.036. 8. So the square root of 0.036 is approximately between 0.18 and 0.19. 9. Since $0.19^2 = 0.0361$ is closer, the estimate is about $0.19$. 10. Alternatively, write $0.036 = 36 \times 10^{-3} = (6 \times 10^{-2})^2 = 0.06^2$ is incorrect, so check carefully. 11. Actually, $0.06^2 = 0.0036$, so $\sqrt{0.036} = \sqrt{36 \times 10^{-3}} = \sqrt{36} \times \sqrt{10^{-3}} = 6 \times 10^{-1.5} = 6 \times 10^{-1} \times 10^{-0.5} = 6 \times 0.1 \times 0.3162 = 6 \times 0.03162 = 0.1897$ approximately. 12. Therefore, $\sqrt{0.036} \approx 0.19$. Final answer: $\boxed{0.19}$