If ` x = 19 and y = 18` then the value of ` ( x^(2) + y^(2) + xy)/( x^(3) - y^(3))` is

Options

A. 1
B. 37
C. 324
D. 361

Answer

To solve the expression \(( x^{2} + y^{2} + xy)/( x^{3} - y^{3})\) given \( x = 19\) and \( y = 18\), we can follow these steps:

### Step 1: Substitute the values of \(x\) and \(y\)
We start by substituting the values of \(x\) and \(y\) into the expression:
\[ x = 19, \quad y = 18 \]
So the expression becomes:
\[ \frac{(19^{2} + 18^{2} + 19 \cdot 18)}{(19^{3} - 18^{3})} \]

### Step 2: Calculate \(x^2\), \(y^2\), and \(xy\)
Now we calculate \(19^{2}\), \(18^{2}\), and \(19 \cdot 18\):
\[ 19^{2} = 361, \quad 18^{2} = 324, \quad 19 \cdot 18 = 342 \]
Now we can add these values together:
\[ 19^{2} + 18^{2} + 19 \cdot 18 = 361 + 324 + 342 = 1027 \]

### Step 3: Calculate \(x^3\) and \(y^3\)
Next, we calculate \(19^{3}\) and \(18^{3}\):
\[ 19^{3} = 6859, \quad 18^{3} = 5832 \]
Now we subtract these values:
\[ 19^{3} - 18^{3} = 6859 - 5832 = 1027 \]

### Step 4: Substitute back into the expression
Now we substitute our results back into the expression:
\[ \frac{1027}{1027} \]

### Step 5: Simplify the expression
This simplifies to:
\[ 1 \]

### Final Answer
Thus, the value of \(\frac{( x^{2} + y^{2} + xy)}{( x^{3} - y^{3})}\) is:
\[ \boxed{1} \]