The largest among the numbers `0.9`, `(0.9)^(2)`, `sqrt(0.9)`, `0.bar9` is

Options

A. `0.9`
B. `(0.9)^(2)`
C. `sqrt(0.9)`
D. `0.bar9`

Answer

To find the largest among the numbers \(0.9\), \((0.9)^2\), \(\sqrt{0.9}\), and \(0.\overline{9}\), we will evaluate each of these numbers step by step.

### Step 1: Evaluate \(0.9\)
The first number is simply \(0.9\).

### Step 2: Evaluate \((0.9)^2\)
Next, we calculate \((0.9)^2\):
\[ (0.9)^2 = 0.9 \times 0.9 = 0.81 \]

### Step 3: Evaluate \(\sqrt{0.9}\)
Now we find the square root of \(0.9\):
\[ \sqrt{0.9} \approx 0.9487 \quad (\text{using a calculator or approximation}) \]

### Step 4: Evaluate \(0.\overline{9}\)
The number \(0.\overline{9}\) represents a repeating decimal, which is equal to \(1\):
\[ 0.\overline{9} = 1 \]

### Step 5: Compare the values
Now we have the following values:
- \(0.9\)
- \((0.9)^2 = 0.81\)
- \(\sqrt{0.9} \approx 0.9487\)
- \(0.\overline{9} = 1\)

### Step 6: Identify the largest number
Comparing these values:
- \(0.9\) is less than \(1\)
- \(0.81\) is less than \(1\)
- \(0.9487\) is less than \(1\)

Thus, the largest number among them is:
\[ \boxed{0.\overline{9}} \]

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