The square root of `0.bar(4)` is

Options

A. `0.bar(6)`
B. `0.bar(7)`
C. `0.bar(8)`
D. `0.bar(9)`

Answer

To find the square root of \(0.\overline{4}\), we will follow these steps:

### Step 1: Convert \(0.\overline{4}\) to a Fraction
The repeating decimal \(0.\overline{4}\) can be expressed as a fraction. For a decimal of the form \(0.ab\overline{c}\), we can use the formula:
\[ 0.ab\overline{c} = \frac{ab}{99} \]
In this case, \(ab = 0.4\), so we can write:
\[ 0.\overline{4} = \frac{4}{9} \]

### Step 2: Find the Square Root of the Fraction
Now we need to find the square root of \(\frac{4}{9}\):
\[ \sqrt{0.\overline{4}} = \sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \]

### Step 3: Convert the Result Back to Decimal
The fraction \(\frac{2}{3}\) can be converted back to a decimal:
\[ \frac{2}{3} = 0.6666\ldots = 0.\overline{6} \]

### Final Answer
Thus, the square root of \(0.\overline{4}\) is:
\[ \sqrt{0.\overline{4}} = 0.\overline{6} \]

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