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 start with the given value of \( x \):

\[ x = 13 + 2\sqrt{42} \]

We need to find \( \sqrt{x} + \frac{1}{\sqrt{x}} \) and express it in the form \( a\sqrt{b} \).

### Step 1: Express \( x \) as a perfect square

We can rewrite \( x \) in a form that resembles a perfect square. We notice that:

\[ x = 13 + 2\sqrt{42} \]

We can express \( 13 \) as \( 6 + 7 \) to facilitate completing the square:

\[ x = (6 + 7) + 2\sqrt{42} \]

Now, we can use the identity \( (a + b)^2 = a^2 + b^2 + 2ab \). We can set \( a = \sqrt{6} \) and \( b = \sqrt{7} \):

\[ x = (\sqrt{6} + \sqrt{7})^2 \]

### Step 2: Calculate \( \sqrt{x} \)

Now that we have expressed \( x \) as a perfect square, we can find \( \sqrt{x} \):

\[ \sqrt{x} = \sqrt{(\sqrt{6} + \sqrt{7})^2} = \sqrt{6} + \sqrt{7} \]

### Step 3: Calculate \( \frac{1}{\sqrt{x}} \)

Next, we need to find \( \frac{1}{\sqrt{x}} \):

\[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{6} + \sqrt{7}} \]

To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{7} - \sqrt{6} \):

\[ \frac{1}{\sqrt{6} + \sqrt{7}} \cdot \frac{\sqrt{7} - \sqrt{6}}{\sqrt{7} - \sqrt{6}} = \frac{\sqrt{7} - \sqrt{6}}{(\sqrt{7})^2 - (\sqrt{6})^2} = \frac{\sqrt{7} - \sqrt{6}}{7 - 6} = \sqrt{7} - \sqrt{6} \]

### Step 4: Combine \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \)

Now we can combine \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \):

\[ \sqrt{x} + \frac{1}{\sqrt{x}} = (\sqrt{6} + \sqrt{7}) + (\sqrt{7} - \sqrt{6}) = 2\sqrt{7} \]

### Step 5: Identify \( a \) and \( b \)

We have expressed \( \sqrt{x} + \frac{1}{\sqrt{x}} \) as:

\[ 2\sqrt{7} = a\sqrt{b} \]

From this, we can identify \( a = 2 \) and \( b = 7 \).

### Step 6: Calculate \( b - a \)

Finally, we need to find \( b - a \):

\[ b - a = 7 - 2 = 5 \]

Thus, the final answer is:

\[ \boxed{5} \]