If `a= 5, b= 4, c=8` then find the value of `(a^(3) + b^(3) + c^(3)-3abc)//(ab+ bc+ ca- a^(2) -b^(2)-c^(2))`
A 15 B 17 C `-17` D `-15`

Options

A. B
B. C
C. D
D. A

Answer

To solve the expression \((a^3 + b^3 + c^3 - 3abc) / (ab + bc + ca - a^2 - b^2 - c^2)\) given \(a = 5\), \(b = 4\), and \(c = 8\), we can follow these steps:

### Step 1: Calculate \(a^3\), \(b^3\), and \(c^3\)
- \(a^3 = 5^3 = 125\)
- \(b^3 = 4^3 = 64\)
- \(c^3 = 8^3 = 512\)

### Step 2: Calculate \(3abc\)
- \(abc = 5 \times 4 \times 8 = 160\)
- Thus, \(3abc = 3 \times 160 = 480\)

### Step 3: Substitute into the numerator
- The numerator becomes:
\[ a^3 + b^3 + c^3 - 3abc = 125 + 64 + 512 - 480 \]
- Calculate:
\[ 125 + 64 = 189 \]
\[ 189 + 512 = 701 \]
\[ 701 - 480 = 221 \]

### Step 4: Calculate \(ab\), \(bc\), and \(ca\)
- \(ab = 5 \times 4 = 20\)
- \(bc = 4 \times 8 = 32\)
- \(ca = 8 \times 5 = 40\)

### Step 5: Calculate \(a^2\), \(b^2\), and \(c^2\)
- \(a^2 = 5^2 = 25\)
- \(b^2 = 4^2 = 16\)
- \(c^2 = 8^2 = 64\)

### Step 6: Substitute into the denominator
- The denominator becomes:
\[ ab + bc + ca - a^2 - b^2 - c^2 = 20 + 32 + 40 - 25 - 16 - 64 \]
- Calculate:
\[ 20 + 32 = 52 \]
\[ 52 + 40 = 92 \]
\[ 92 - 25 = 67 \]
\[ 67 - 16 = 51 \]
\[ 51 - 64 = -13 \]

### Step 7: Final calculation of the expression
- Now we can substitute the values into the expression:
\[ \frac{221}{-13} = -17 \]

### Final Answer
Thus, the value of the expression is \(-17\).