If `x+ (1)/(x)=3`, then
`x^(3) + (1)/(x^(3))`=?

Answer

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

### Step 1: Cube both sides of the equation
We start with the equation:
\[ x + \frac{1}{x} = 3 \]
Now, we will cube both sides:
\[ \left( x + \frac{1}{x} \right)^3 = 3^3 \]
This simplifies to:
\[ x^3 + 3x\left(\frac{1}{x}\right)(x + \frac{1}{x}) + \frac{1}{x^3} = 27 \]

### Step 2: Simplify the equation
We know that:
\[ 3x\left(\frac{1}{x}\right) = 3 \]
Thus, substituting this into the equation gives us:
\[ x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}) = 27 \]
Since \( x + \frac{1}{x} = 3 \), we can substitute this into the equation:
\[ x^3 + \frac{1}{x^3} + 3(3) = 27 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 9 = 27 \]

### Step 3: Isolate \( x^3 + \frac{1}{x^3} \)
Now, we can isolate \( x^3 + \frac{1}{x^3} \):
\[ x^3 + \frac{1}{x^3} = 27 - 9 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} = 18 \]

### Final Answer
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{18} \]