If a= 25, b= 15, `c= - 10` then `(a^(3) + b^(3) + c^(3) - 3abc)/((a-b)^(2) + (b-c)^(2) + (c-a)^(2))`

Answer

To solve the expression

\[ \frac{a^3 + b^3 + c^3 - 3abc}{(a-b)^2 + (b-c)^2 + (c-a)^2} \]

given \( a = 25 \), \( b = 15 \), and \( c = -10 \), we can follow these steps:

### Step 1: Calculate \( a^3 + b^3 + c^3 \)

First, we calculate \( a^3 \), \( b^3 \), and \( c^3 \):

\[ a^3 = 25^3 = 15625 \]
\[ b^3 = 15^3 = 3375 \]
\[ c^3 = (-10)^3 = -1000 \]

Now, sum these values:

\[ a^3 + b^3 + c^3 = 15625 + 3375 - 1000 = 17900 \]

### Step 2: Calculate \( 3abc \)

Next, we calculate \( 3abc \):

\[ abc = 25 \times 15 \times (-10) = -3750 \]
\[ 3abc = 3 \times (-3750) = -11250 \]

### Step 3: Substitute into the numerator

Now substitute into the numerator:

\[ a^3 + b^3 + c^3 - 3abc = 17900 - (-11250) = 17900 + 11250 = 29150 \]

### Step 4: Calculate the denominator

Now we calculate the denominator:

\[ (a-b)^2 = (25 - 15)^2 = 10^2 = 100 \]
\[ (b-c)^2 = (15 - (-10))^2 = (15 + 10)^2 = 25^2 = 625 \]
\[ (c-a)^2 = (-10 - 25)^2 = (-35)^2 = 1225 \]

Now sum these values:

\[ (a-b)^2 + (b-c)^2 + (c-a)^2 = 100 + 625 + 1225 = 1950 \]

### Step 5: Combine the results

Now we can combine the results into the final expression:

\[ \frac{29150}{1950} \]

### Step 6: Simplify the fraction

Now simplify the fraction:

\[ \frac{29150 \div 1950}{1950 \div 1950} = \frac{15}{1} = 15 \]

### Final Answer

Thus, the value of the expression is

\[ \boxed{15} \]