If x + y = p and xy = q, then find the value of `(1)/(x^(3)) + (1)/(y^(3))` will be

Options

A. `(p^(3) -3)/(q^(2))`
B. `(p^(3)-3p)/(q^(3))`
C. `(p^(3)-3pq)/(q^(3))`
D. `(q^(3) - 3pq)/(p^(2)`

Answer

To find the value of \( \frac{1}{x^3} + \frac{1}{y^3} \) given that \( x + y = p \) and \( xy = q \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression we want to find:
\[ \frac{1}{x^3} + \frac{1}{y^3} \]
This can be rewritten using a common denominator:
\[ \frac{y^3 + x^3}{x^3 y^3} \]

### Step 2: Use the identity for \( x^3 + y^3 \)
We can use the identity:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We already know that \( x + y = p \). To find \( x^2 - xy + y^2 \), we can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy = p^2 - 2q \]
Thus, we have:
\[ x^2 - xy + y^2 = (x^2 + y^2) - xy = (p^2 - 2q) - q = p^2 - 3q \]

### Step 3: Substitute back into the expression
Now substituting back into the expression for \( x^3 + y^3 \):
\[ x^3 + y^3 = p(p^2 - 3q) \]
So we can rewrite our original expression:
\[ \frac{1}{x^3} + \frac{1}{y^3} = \frac{x^3 + y^3}{(xy)^3} = \frac{p(p^2 - 3q)}{(xy)^3} = \frac{p(p^2 - 3q)}{q^3} \]

### Final Expression
Thus, we arrive at the final result:
\[ \frac{1}{x^3} + \frac{1}{y^3} = \frac{p(p^2 - 3q)}{q^3} \]