If a= 20, b= 25, c= 15 find `(a^(3) +b^(3) + c^(3)- 3 abc)/(a^(2) + b^(2) + c^(2) - ab- bc - ca)`= ?

Answer

To solve the problem, we need to evaluate the expression:

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

Given values are:
- \( a = 20 \)
- \( b = 25 \)
- \( c = 15 \)

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

First, we find the sum of \( a \), \( b \), and \( c \):

\[ a + b + c = 20 + 25 + 15 \]

Calculating this gives:

\[ a + b + c = 60 \]

### Step 2: Calculate \( a^2 + b^2 + c^2 \)

Next, we calculate \( a^2 + b^2 + c^2 \):

\[ a^2 = 20^2 = 400 \]
\[ b^2 = 25^2 = 625 \]
\[ c^2 = 15^2 = 225 \]

Now, summing these:

\[ a^2 + b^2 + c^2 = 400 + 625 + 225 = 1250 \]

### Step 3: Calculate \( ab + bc + ca \)

Now we calculate \( ab + bc + ca \):

\[ ab = 20 \times 25 = 500 \]
\[ bc = 25 \times 15 = 375 \]
\[ ca = 15 \times 20 = 300 \]

Now, summing these:

\[ ab + bc + ca = 500 + 375 + 300 = 1175 \]

### Step 4: Calculate \( a^2 + b^2 + c^2 - ab - bc - ca \)

Now we can find:

\[ a^2 + b^2 + c^2 - ab - bc - ca = 1250 - 1175 = 75 \]

### Step 5: Use the identity for \( a^3 + b^3 + c^3 - 3abc \)

Using the identity:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \]

Substituting the values we calculated:

\[ a^3 + b^3 + c^3 - 3abc = (60)(75) = 4500 \]

### Step 6: Substitute into the original expression

Now we substitute into the original expression:

\[ \frac{4500}{75} \]

Calculating this gives:

\[ \frac{4500}{75} = 60 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{60} \]