If `sqrtx+(1)/(sqrtx)=3`, then the value of `x^3+(1)/(x^3)` is:

Options

A. 322
B. 326
C. 324
D. 422

Answer

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

### Step 1: Let \( y = \sqrt{x} \)
We can rewrite the equation as:
\[ y + \frac{1}{y} = 3 \]

### Step 2: Square both sides
Squaring both sides gives us:
\[ \left( y + \frac{1}{y} \right)^2 = 3^2 \]
This simplifies to:
\[ y^2 + 2 + \frac{1}{y^2} = 9 \]

### Step 3: Rearrange the equation
Rearranging the equation, we get:
\[ y^2 + \frac{1}{y^2} = 9 - 2 = 7 \]

### Step 4: Find \( y^3 + \frac{1}{y^3} \)
We can use the identity:
\[ y^3 + \frac{1}{y^3} = \left( y + \frac{1}{y} \right) \left( y^2 + \frac{1}{y^2} \right) - \left( y + \frac{1}{y} \right) \]
Substituting the known values:
\[ y^3 + \frac{1}{y^3} = 3 \cdot 7 - 3 \]

### Step 5: Calculate the result
Calculating this gives:
\[ y^3 + \frac{1}{y^3} = 21 - 3 = 18 \]

### Step 6: Conclusion
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{18} \]