Which of the following numbers `0.1 , 0.11 ,(0.11)^(2), sqrt(0.001)` is the greatest ?

Answer

To determine which of the following numbers is the greatest: \(0.1\), \(0.11\), \((0.11)^2\), and \(\sqrt{0.001}\), we can evaluate each number step by step.

### Step 1: Compare \(0.1\) and \(0.11\)
- We can express \(0.1\) as \(0.10\) to make the comparison easier.
- Clearly, \(0.10 < 0.11\).

**Hint:** When comparing decimals, look at the digits from left to right. The first digit that differs determines which number is greater.

### Step 2: Calculate \((0.11)^2\)
- To find \((0.11)^2\), we multiply \(0.11\) by itself:
\[ (0.11) \times (0.11) = 0.0121 \]

**Hint:** Squaring a decimal less than 1 will yield a smaller number. Remember to multiply the decimal places correctly.

### Step 3: Calculate \(\sqrt{0.001}\)
- We can rewrite \(0.001\) as \(\frac{1}{1000}\).
- The square root of \(0.001\) is:
\[ \sqrt{0.001} = \sqrt{\frac{1}{1000}} = \frac{\sqrt{1}}{\sqrt{1000}} = \frac{1}{\sqrt{1000}} \]
- Now, \(\sqrt{1000}\) can be approximated. Since \(31.62^2 \approx 1000\), we have:
\[ \sqrt{1000} \approx 31.62 \implies \frac{1}{\sqrt{1000}} \approx \frac{1}{31.62} \approx 0.0316 \]

**Hint:** When calculating square roots, remember that \(\sqrt{a/b} = \frac{\sqrt{a}}{\sqrt{b}}\).

### Step 4: Compare all values
- Now we have:
- \(0.1\) (which is \(0.10\))
- \(0.11\)
- \((0.11)^2 = 0.0121\)
- \(\sqrt{0.001} \approx 0.0316\)

**Step 5: Determine the greatest number**
- Comparing \(0.1\), \(0.11\), \(0.0121\), and \(0.0316\):
- \(0.11\) is greater than \(0.1\)
- \(0.11\) is greater than \(0.0316\)
- \(0.11\) is greater than \(0.0121\)

Thus, the greatest number among the given options is \(0.11\).

**Final Answer:** The greatest number is \(0.11\).