Add: `(7)/((-18))and(8)/(27)`

Answer

To solve the problem of adding the rational numbers \( \frac{7}{-18} \) and \( \frac{8}{27} \), we can follow these steps:

### Step 1: Rewrite the first fraction
The first fraction is \( \frac{7}{-18} \). We can rewrite it as:
\[ -\frac{7}{18} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators 18 and 27.

- The prime factorization of 18 is \( 2 \times 3^2 \).
- The prime factorization of 27 is \( 3^3 \).

To find the LCM, we take the highest power of each prime factor:
- For 2: \( 2^1 \)
- For 3: \( 3^3 \)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^3 = 2 \times 27 = 54 \]

### Step 3: Convert each fraction to have the LCM as the denominator
Now we convert each fraction to have a denominator of 54.

For \( -\frac{7}{18} \):
\[ -\frac{7}{18} = -\frac{7 \times 3}{18 \times 3} = -\frac{21}{54} \]

For \( \frac{8}{27} \):
\[ \frac{8}{27} = \frac{8 \times 2}{27 \times 2} = \frac{16}{54} \]

### Step 4: Add the fractions
Now we can add the two fractions:
\[ -\frac{21}{54} + \frac{16}{54} = \frac{-21 + 16}{54} = \frac{-5}{54} \]

### Final Answer
Thus, the final answer is:
\[ \frac{-5}{54} \]

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