If `a+(1)/(a)=1` then what is the value of `a^(3)`?

Options

A. `-2`
B. 2
C. `-1`
D. 4

Answer

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

### Step 1: Start with the given equation
We have:
\[ a + \frac{1}{a} = 1 \]

### Step 2: Cube both sides
Cubing both sides of the equation gives us:
\[ \left(a + \frac{1}{a}\right)^3 = 1^3 \]
This expands to:
\[ a^3 + 3a\left(\frac{1}{a}\right)(a + \frac{1}{a}) + \frac{1}{a^3} = 1 \]
Simplifying the middle term:
\[ 3a\left(\frac{1}{a}\right)(a + \frac{1}{a}) = 3(a + \frac{1}{a}) = 3 \cdot 1 = 3 \]
Thus, we have:
\[ a^3 + \frac{1}{a^3} + 3 = 1 \]

### Step 3: Rearranging the equation
Now, we can rearrange the equation:
\[ a^3 + \frac{1}{a^3} = 1 - 3 \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} = -2 \]

### Step 4: Express \( a^3 \)
Let \( x = a^3 \). Then we can write:
\[ x + \frac{1}{x} = -2 \]

### Step 5: Multiply through by \( x \)
Multiplying both sides by \( x \) gives:
\[ x^2 + 1 = -2x \]
Rearranging this gives us a quadratic equation:
\[ x^2 + 2x + 1 = 0 \]

### Step 6: Factor the quadratic
This can be factored as:
\[ (x + 1)^2 = 0 \]
Thus, we find:
\[ x + 1 = 0 \implies x = -1 \]

### Step 7: Conclusion
Since \( x = a^3 \), we have:
\[ a^3 = -1 \]

### Final Answer
The value of \( a^3 \) is:
\[ \boxed{-1} \]