Solve the following expression :
`193/75 + 37/81 - 81/65`.

Answer

To solve the expression \( \frac{193}{75} + \frac{37}{81} - \frac{81}{65} \), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the denominators
The denominators are 75, 81, and 65. We need to find the prime factorization of each:

- \( 75 = 3 \times 5^2 \)
- \( 81 = 3^4 \)
- \( 65 = 5 \times 13 \)

Now, we take the highest power of each prime factor:
- From \( 3 \): highest power is \( 3^4 \) (from 81)
- From \( 5 \): highest power is \( 5^2 \) (from 75)
- From \( 13 \): highest power is \( 13^1 \) (from 65)

Thus, the LCM is:
\[ LCM = 3^4 \times 5^2 \times 13 = 81 \times 25 \times 13 = 26325 \]

### Step 2: Rewrite each fraction with the common denominator
Now we will convert each fraction to have the denominator of 26325.

1. For \( \frac{193}{75} \):
\[ \frac{193}{75} = \frac{193 \times 351}{75 \times 351} = \frac{67843}{26325} \]

2. For \( \frac{37}{81} \):
\[ \frac{37}{81} = \frac{37 \times 325}{81 \times 325} = \frac{12025}{26325} \]

3. For \( \frac{81}{65} \):
\[ \frac{81}{65} = \frac{81 \times 405}{65 \times 405} = \frac{32805}{26325} \]

### Step 3: Combine the fractions
Now we can combine the fractions:
\[ \frac{67843}{26325} + \frac{12025}{26325} - \frac{32805}{26325} = \frac{67843 + 12025 - 32805}{26325} \]

Calculating the numerator:
\[ 67843 + 12025 = 79868 \]
\[ 79868 - 32805 = 47063 \]

Thus, we have:
\[ \frac{47063}{26325} \]

### Step 4: Simplify the fraction if possible
To check if the fraction can be simplified, we find the GCD of 47063 and 26325. Since they do not have common factors, we can conclude that \( \frac{47063}{26325} \) is in its simplest form.

### Final Answer:
\[ \frac{47063}{26325} \]

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