If `(a)/(b)+(b)/(a)= -1` then `(a^3-b^3)=?`

Options

A. `-3`
B. `-2`
C. `-1`
D. `0`

Explanation

To solve the equation \(\frac{a}{b} + \frac{b}{a} = -1\) and find the value of \(a^3 - b^3\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{a}{b} + \frac{b}{a} = -1 \]
To combine the fractions, we find a common denominator, which is \(ab\):
\[ \frac{a^2 + b^2}{ab} = -1 \]

### Step 2: Clear the fraction
Multiply both sides by \(ab\) to eliminate the denominator:
\[ a^2 + b^2 = -ab \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ a^2 + b^2 + ab = 0 \]

### Step 4: Use the identity for \(a^3 - b^3\)
We know the formula for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + b^2 + ab) \]
From Step 3, we have \(a^2 + b^2 + ab = 0\).

### Step 5: Substitute into the formula
Substituting this into the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(0) \]

### Step 6: Simplify the expression
Since anything multiplied by zero is zero, we find:
\[ a^3 - b^3 = 0 \]

### Final Answer:
Thus, the value of \(a^3 - b^3\) is:
\[ \boxed{0} \]

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