Value of `(a^(3)+b^(3)+c^(3)-3abc)/(ab+bc+ca-a^(2)-b^(2)-c^(2))` , when `a=-5, b=-6, c=10` is

Answer

To find the value of the expression

\[ \frac{a^3 + b^3 + c^3 - 3abc}{ab + bc + ca - a^2 - b^2 - c^2} \]

when \( a = -5 \), \( b = -6 \), and \( c = 10 \), we can follow these steps:

### Step 1: Calculate \( a^3 + b^3 + c^3 \)
First, we calculate each cube:
- \( a^3 = (-5)^3 = -125 \)
- \( b^3 = (-6)^3 = -216 \)
- \( c^3 = (10)^3 = 1000 \)

Now, sum these values:
\[ a^3 + b^3 + c^3 = -125 - 216 + 1000 = 659 \]

### Step 2: Calculate \( 3abc \)
Next, we calculate \( 3abc \):
\[ abc = (-5) \cdot (-6) \cdot (10) = 300 \]
Thus,
\[ 3abc = 3 \cdot 300 = 900 \]

### Step 3: Calculate the numerator
Now we can find the numerator:
\[ a^3 + b^3 + c^3 - 3abc = 659 - 900 = -241 \]

### Step 4: Calculate \( ab + bc + ca \)
Now, we calculate each product:
- \( ab = (-5) \cdot (-6) = 30 \)
- \( bc = (-6) \cdot (10) = -60 \)
- \( ca = (10) \cdot (-5) = -50 \)

Now, sum these values:
\[ ab + bc + ca = 30 - 60 - 50 = -80 \]

### Step 5: Calculate \( a^2 + b^2 + c^2 \)
Next, we calculate the squares:
- \( a^2 = (-5)^2 = 25 \)
- \( b^2 = (-6)^2 = 36 \)
- \( c^2 = (10)^2 = 100 \)

Now, sum these values:
\[ a^2 + b^2 + c^2 = 25 + 36 + 100 = 161 \]

### Step 6: Calculate the denominator
Now we can find the denominator:
\[ ab + bc + ca - a^2 - b^2 - c^2 = -80 - 161 = -241 \]

### Step 7: Calculate the final value
Now we can substitute the values into the original expression:
\[ \frac{-241}{-241} = 1 \]

### Final Answer
The value of the expression is \( 1 \).

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