`(a)/(b) + (b)/(a) =1,` then the value of `a ^(3) + b ^(3)` is

Options

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

Answer

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 given equation
We start with the equation:
\[ \frac{a}{b} + \frac{b}{a} = 1 \]
We can combine the fractions on the left side:
\[ \frac{a^2 + b^2}{ab} = 1 \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ a^2 + b^2 = ab \]

### Step 3: Use the identity for \(a^3 + b^3\)
We use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
From Step 2, we know \(a^2 + b^2 = ab\). Thus, we can substitute this into the identity:
\[ a^3 + b^3 = (a + b)(ab - ab) = (a + b)(0) \]

### Step 4: Conclude the value of \(a^3 + b^3\)
Since we have \((a + b)(0) = 0\), we find that:
\[ a^3 + b^3 = 0 \]

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

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