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

Options

A. `0.bar(8)`
B. `0.bar(6)`
C. `0.bar(7)`
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
Let \(x = 0.\overline{4}\). This means that \(x = 0.4444...\)

To convert this repeating decimal into a fraction, we can use the following method:

1. Multiply \(x\) by 10 to shift the decimal point:
\[ 10x = 4.4444... \]

2. Now, subtract the original \(x\) from this equation:
\[ 10x - x = 4.4444... - 0.4444... \]
\[ 9x = 4 \]

3. Solve for \(x\):
\[ x = \frac{4}{9} \]

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

### Step 3: Convert the Result to Decimal Form
Now, we can convert \(\frac{2}{3}\) into decimal form:
\[ \frac{2}{3} = 0.6666... = 0.\overline{6} \]

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

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