Find the value of `3.bar4xx5.bar6`

Answer

To find the value of \(3.\overline{4} \times 5.\overline{6}\), we will follow these steps:

### Step 1: Define the repeating decimals
Let:
- \(a = 3.\overline{4}\)
- \(b = 5.\overline{6}\)

### Step 2: Convert \(a\) into a fraction
To convert \(3.\overline{4}\) into a fraction:
1. Write \(a = 3.4444...\)
2. Multiply both sides by 10 to shift the decimal point:
\[ 10a = 34.4444... \]
3. Now, subtract the original \(a\) from this equation:
\[ 10a - a = 34.4444... - 3.4444... \]
This simplifies to:
\[ 9a = 31 \]
4. Solve for \(a\):
\[ a = \frac{31}{9} \]

### Step 3: Convert \(b\) into a fraction
To convert \(5.\overline{6}\) into a fraction:
1. Write \(b = 5.6666...\)
2. Multiply both sides by 10:
\[ 10b = 56.6666... \]
3. Subtract the original \(b\) from this equation:
\[ 10b - b = 56.6666... - 5.6666... \]
This simplifies to:
\[ 9b = 51 \]
4. Solve for \(b\):
\[ b = \frac{51}{9} \]

### Step 4: Multiply the fractions
Now we will multiply \(a\) and \(b\):
\[ 3.\overline{4} \times 5.\overline{6} = a \times b = \frac{31}{9} \times \frac{51}{9} = \frac{31 \times 51}{9 \times 9} \]

### Step 5: Calculate the numerator
Now we calculate \(31 \times 51\):
\[ 31 \times 51 = 1581 \]
So, we have:
\[ \frac{1581}{81} \]

### Step 6: Simplify the fraction
Now we will simplify \(\frac{1581}{81}\):
1. Divide both the numerator and the denominator by 3:
\[ \frac{1581 \div 3}{81 \div 3} = \frac{527}{27} \]

### Step 7: Convert to mixed number
To convert \(\frac{527}{27}\) to a mixed number:
1. Divide \(527\) by \(27\):
- \(27\) goes into \(527\) \(19\) times (since \(27 \times 19 = 513\)).
- The remainder is \(527 - 513 = 14\).
2. Thus, we can write:
\[ \frac{527}{27} = 19 \frac{14}{27} \]

### Final Answer
Thus, the value of \(3.\overline{4} \times 5.\overline{6}\) is:
\[ 19 \frac{14}{27} \]