`5.7bar(32) + 8.bar(613)`

Answer

To solve the problem \( 5.7\overline{32} + 8.\overline{613} \), we need to first convert the repeating decimals into fractions and then perform the addition.

### Step 1: Convert \( 5.7\overline{32} \) into a fraction

Let \( x = 5.7\overline{32} \).

To eliminate the repeating part, we can express it as follows:

1. Multiply \( x \) by \( 1000 \) (to shift the decimal point three places to the right):
\[ 1000x = 5732.\overline{32} \]

2. Multiply \( x \) by \( 10 \) (to shift the decimal point one place to the right):
\[ 10x = 57.\overline{32} \]

3. Now, subtract the second equation from the first:
\[ 1000x - 10x = 5732.\overline{32} - 57.\overline{32} \]
\[ 990x = 5732 - 57 \]
\[ 990x = 5675 \]
\[ x = \frac{5675}{990} \]

4. Simplify \( \frac{5675}{990} \):
- The GCD of 5675 and 990 is 5.
\[ x = \frac{5675 \div 5}{990 \div 5} = \frac{1135}{198} \]

### Step 2: Convert \( 8.\overline{613} \) into a fraction

Let \( y = 8.\overline{613} \).

1. Multiply \( y \) by \( 1000 \):
\[ 1000y = 8613.\overline{613} \]

2. Multiply \( y \) by \( 10 \):
\[ 10y = 86.\overline{613} \]

3. Subtract the second equation from the first:
\[ 1000y - 10y = 8613.\overline{613} - 86.\overline{613} \]
\[ 990y = 8613 - 86 \]
\[ 990y = 8527 \]
\[ y = \frac{8527}{990} \]

### Step 3: Add the two fractions

Now we have:
\[ x = \frac{1135}{198}, \quad y = \frac{8527}{990} \]

To add \( x \) and \( y \), we need a common denominator. The least common multiple (LCM) of 198 and 990 is 990.

1. Convert \( x \) to have a denominator of 990:
\[ x = \frac{1135 \times 5}{198 \times 5} = \frac{5675}{990} \]

2. Now add \( x \) and \( y \):
\[ x + y = \frac{5675}{990} + \frac{8527}{990} = \frac{5675 + 8527}{990} = \frac{14202}{990} \]

3. Simplify \( \frac{14202}{990} \):
- The GCD of 14202 and 990 is 6.
\[ \frac{14202 \div 6}{990 \div 6} = \frac{2367}{165} \]

### Final Answer
Thus, the final result of \( 5.7\overline{32} + 8.\overline{613} \) is:
\[ \frac{2367}{165} \]