Which one of the following numbers has rational square root?

### Concept & Strategy For a decimal number to have a rational square root, when converted to a fraction of the form $p/q$, both the numerator and the denominator (after simplification or scaling to a power of $100$) must be perfect squares. ### Step-by-Step Solution Convert each decimal option into a fraction to inspect its square root: * Option (a) $0.4 = \frac{4}{10}$. Taking the square root gives $\frac{2}{\sqrt{10}}$, which is irrational. * Option (b) $0.09 = \frac{9}{100}$. Taking the square root gives $\frac{3}{10} = 0.3$, which is a rational number. * Option (c) $0.9 = \frac{9}{10}$. Taking the square root gives $\frac{3}{\sqrt{10}}$, which is irrational. * Option (d) $0.025 = \frac{25}{1000} = \frac{1}{40}$. Taking the square root gives $\frac{1}{\sqrt{40}}$, which is irrational. ### Exam Strategy & Shortcut Count the decimal places! For a base-10 decimal to be a perfect square, it MUST have an even number of decimal places (e.g., 2, 4, 6). * $0.4$, $0.9$, and $0.025$ have odd numbers of decimal places ($1$, $1$, and $3$ respectively). They cannot be perfect squares. * $0.09$ has $2$ decimal places and $9$ is a perfect square. Thus, it is the only viable candidate. ### Common Pitfall Students often see $4$, $9$, and $25$ and mistakenly assume all these options yield rational roots. Always check the decimal place count; an odd number of decimal places always results in an irrational square root involving $\sqrt{10}$. ### Final Answer **Therefore, the correct answer is 0.09.**