Determine the value of `((1)/( r) + (1)/( s)) " when " r^(3) + s^(3) = 0 ` and r + s = 6

Options

A. 0
B. `0.5`
C. 1
D. 6

Answer

To solve the problem of determining the value of \(\frac{1}{r} + \frac{1}{s}\) given that \(r^3 + s^3 = 0\) and \(r + s = 6\), we can follow these steps:

### Step 1: Use the identity for the sum of cubes
We know that:
\[ r^3 + s^3 = (r + s)(r^2 - rs + s^2) \]
Given that \(r^3 + s^3 = 0\), we can set up the equation:
\[ 0 = (r + s)(r^2 - rs + s^2) \]

### Step 2: Substitute the known value of \(r + s\)
Since we know \(r + s = 6\), we can substitute this into the equation:
\[ 0 = 6(r^2 - rs + s^2) \]
This implies:
\[ r^2 - rs + s^2 = 0 \]

### Step 3: Use the identity for \(r^2 + s^2\)
We can express \(r^2 + s^2\) in terms of \(r + s\) and \(rs\):
\[ r^2 + s^2 = (r + s)^2 - 2rs \]
Substituting \(r + s = 6\):
\[ r^2 + s^2 = 6^2 - 2rs = 36 - 2rs \]
Now, substituting this into the equation from Step 2:
\[ 36 - 2rs - rs = 0 \]
This simplifies to:
\[ 36 - 3rs = 0 \]

### Step 4: Solve for \(rs\)
Rearranging gives:
\[ 3rs = 36 \implies rs = \frac{36}{3} = 12 \]

### Step 5: Find \(\frac{1}{r} + \frac{1}{s}\)
We can express \(\frac{1}{r} + \frac{1}{s}\) as:
\[ \frac{1}{r} + \frac{1}{s} = \frac{s + r}{rs} \]
Substituting the known values \(r + s = 6\) and \(rs = 12\):
\[ \frac{1}{r} + \frac{1}{s} = \frac{6}{12} = \frac{1}{2} \]

### Final Answer
Thus, the value of \(\frac{1}{r} + \frac{1}{s}\) is \(\frac{1}{2}\).

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