If `x = 13 + 2sqrt(42)`, then `sqrt(x) + (1)/(sqrt(x))` is equal to `asqrt(b)` then find the value of `b - a` ?

Explanation

To solve the problem, we need to find the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) where \( x = 13 + 2\sqrt{42} \).

### Step-by-Step Solution:

1. **Identify \( x \)**:
\[ x = 13 + 2\sqrt{42} \]

2. **Express \( x \) in a perfect square form**:
We can express \( x \) as:
\[ x = (a + b)^2 \]
where \( a^2 + b^2 + 2ab = 13 + 2\sqrt{42} \).

3. **Set up equations**:
From the equation, we can equate:
- \( 2ab = 2\sqrt{42} \) ⇒ \( ab = \sqrt{42} \)
- \( a^2 + b^2 = 13 \)

4. **Use the identity \( (a + b)^2 = a^2 + b^2 + 2ab \)**:
From \( ab = \sqrt{42} \), we can find \( a \) and \( b \) using:
\[ a^2 + b^2 = 13 \quad \text{and} \quad ab = \sqrt{42} \]

5. **Substituting \( b = \frac{\sqrt{42}}{a} \)** into \( a^2 + b^2 = 13 \)**:
\[ a^2 + \left(\frac{\sqrt{42}}{a}\right)^2 = 13 \]
\[ a^2 + \frac{42}{a^2} = 13 \]
Multiply through by \( a^2 \):
\[ a^4 - 13a^2 + 42 = 0 \]
Let \( y = a^2 \):
\[ y^2 - 13y + 42 = 0 \]

6. **Solve the quadratic equation**:
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ y = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 1 \cdot 42}}{2 \cdot 1} \]
\[ y = \frac{13 \pm \sqrt{169 - 168}}{2} \]
\[ y = \frac{13 \pm 1}{2} \]
Thus, \( y = 7 \) or \( y = 6 \).

7. **Find \( a \) and \( b \)**:
If \( a^2 = 7 \) and \( b^2 = 6 \), then:
\[ a = \sqrt{7}, \quad b = \sqrt{6} \quad \text{or vice versa} \]

8. **Calculate \( \sqrt{x} \)**:
\[ \sqrt{x} = a + b = \sqrt{7} + \sqrt{6} \]

9. **Calculate \( \sqrt{x} + \frac{1}{\sqrt{x}} \)**:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{7} + \sqrt{6} + \frac{1}{\sqrt{7} + \sqrt{6}} \]
To simplify \( \frac{1}{\sqrt{7} + \sqrt{6}} \), we rationalize:
\[ \frac{1}{\sqrt{7} + \sqrt{6}} \cdot \frac{\sqrt{7} - \sqrt{6}}{\sqrt{7} - \sqrt{6}} = \frac{\sqrt{7} - \sqrt{6}}{1} = \sqrt{7} - \sqrt{6} \]
Therefore,
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{7} + \sqrt{6} + \sqrt{7} - \sqrt{6} = 2\sqrt{7} \]

10. **Identify \( a \) and \( b \)**:
Here, we can see that \( \sqrt{x} + \frac{1}{\sqrt{x}} = 2\sqrt{7} \) can be expressed as \( a\sqrt{b} \) where \( a = 2 \) and \( b = 7 \).

11. **Calculate \( b - a \)**:
\[ b - a = 7 - 2 = 5 \]

### Final Answer:
\[ \boxed{5} \]