If a= 2, b= 3 and c= 4, find the value of : `(ab + bc + ca- a^(2) -b^(2)-c^(2))/(3abc-a^(3)-b^(3)-c^(3))`, using suitable identity

Answer

To solve the expression \((ab + bc + ca - a^2 - b^2 - c^2) / (3abc - a^3 - b^3 - c^3)\) given \(a = 2\), \(b = 3\), and \(c = 4\), we can use suitable identities.

### Step-by-Step Solution:

1. **Substituting Values**:
Substitute \(a\), \(b\), and \(c\) into the expression.
\[ \text{Numerator: } ab + bc + ca - a^2 - b^2 - c^2 = (2 \cdot 3) + (3 \cdot 4) + (4 \cdot 2) - (2^2) - (3^2) - (4^2) \]
\[ = 6 + 12 + 8 - 4 - 9 - 16 \]
\[ = 26 - 29 = -3 \]

\[ \text{Denominator: } 3abc - a^3 - b^3 - c^3 = 3(2 \cdot 3 \cdot 4) - (2^3) - (3^3) - (4^3) \]
\[ = 3(24) - 8 - 27 - 64 \]
\[ = 72 - 99 = -27 \]

2. **Forming the Expression**:
Now we can write the expression with the calculated values:
\[ \frac{-3}{-27} \]

3. **Simplifying**:
Simplifying the fraction:
\[ = \frac{3}{27} = \frac{1}{9} \]

### Final Answer:
Thus, the value of the expression is \(\frac{1}{9}\).

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