The square root of `0.bar4` is :

Options

A. `0.bar8`
B. `0.bar6`
C. `0.bar7`
D. `0.bar9`

Answer

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

### Step 1: Convert \(0.\overline{4}\) into a Fraction
The repeating decimal \(0.\overline{4}\) can be expressed as a fraction.

Let \(x = 0.\overline{4}\).

To eliminate the repeating part, multiply both sides by 10:
\[ 10x = 4.\overline{4} \]

Now, subtract the first equation from this new equation:
\[ 10x - x = 4.\overline{4} - 0.\overline{4} \]
\[ 9x = 4 \]
\[ x = \frac{4}{9} \]

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

### Step 3: Convert the Result Back to a Repeating Decimal
To express \(\frac{2}{3}\) as a decimal, we can perform the division:
\[ 2 \div 3 = 0.6666...\text{ (which is } 0.\overline{6}\text{)} \]

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

### Summary
The square root of \(0.\overline{4}\) is \(0.\overline{6}\).

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