We are given \( x = 10 + 3\sqrt{11} \) and asked to find \( \sqrt{x} + \frac{1}{\sqrt{x}} \).

### Step 1: Let \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \).
We can start by squaring both sides of the equation for \( y \):
\[
y^2 = \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2
\]
Expanding this:
\[
y^2 = \left( \sqrt{x} \right)^2 + 2 \cdot \sqrt{x} \cdot \frac{1}{\sqrt{x}} + \left( \frac{1}{\sqrt{x}} \right)^2
\]
This simplifies to:
\[
y^2 = x + 2 + \frac{1}{x}
\]
Thus, we have:
\[
y^2 = x + 2 + \frac{1}{x}
\]

### Step 2: Substitute the given value of \( x = 10 + 3\sqrt{11} \).
Next, we substitute \( x = 10 + 3\sqrt{11} \) into the equation \( y^2 = x + 2 + \frac{1}{x} \).
However, before proceeding, we need to compute \( \frac{1}{x} \).

### Step 3: Find \( \frac{1}{x} \).
To find \( \frac{1}{x} \), multiply both the numerator and denominator by the conjugate of \( x = 10 + 3\sqrt{11} \), which is \( 10 - 3\sqrt{11} \):
\[
\frac{1}{x} = \frac{1}{10 + 3\sqrt{11}} \cdot \frac{10 - 3\sqrt{11}}{10 - 3\sqrt{11}} = \frac{10 - 3\sqrt{11}}{(10 + 3\sqrt{11})(10 - 3\sqrt{11})}
\]
The denominator simplifies as follows:
\[
(10 + 3\sqrt{11})(10 - 3\sqrt{11}) = 10^2 - (3\sqrt{11})^2 = 100 - 99 = 1
\]
So:
\[
\frac{1}{x} = 10 - 3\sqrt{11}
\]

### Step 4: Substitute \( x \) and \( \frac{1}{x} \) into \( y^2 \).
Now, substitute \( x = 10 + 3\sqrt{11} \) and \( \frac{1}{x} = 10 - 3\sqrt{11} \) into the equation for \( y^2 \):
\[
y^2 = (10 + 3\sqrt{11}) + 2 + (10 - 3\sqrt{11})
\]
Simplifying:
\[
y^2 = 10 + 3\sqrt{11} + 2 + 10 - 3\sqrt{11} = 22
\]
Thus, \( y^2 = 22 \), so:
\[
y = \sqrt{22}
\]

### Final Answer:
The value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is \( \sqrt{22} \).

Would you like further details or have any questions?

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Here are 5 related questions for practice:
1. If \( x = 16 \), find \( \sqrt{x} + \frac{1}{\sqrt{x}} \).
2. Simplify \( \frac{1}{5 + 2\sqrt{3}} \).
3. Solve for \( y \) if \( y^2 = 25 \).
4. If \( x = 5 + 2\sqrt{6} \), compute \( \frac{1}{x} \).
5. Find the value of \( \sqrt{9 + 4\sqrt{5}} \).

**Tip:** When simplifying radicals, using the conjugate helps remove square roots from the denominator.

Learn how to find √x + 1/√x when x = 10 + 3√11 with this detailed step-by-step solution. Understand key algebraic concepts such as radical expressions, the use of conjugates, and properties of square roots. Suitable for students in Grades 9-11.

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