To simplify the expression $ \sqrt{3 + \sqrt{3}} $, let's analyze it step by step.

### Step 1: Set up the equation
Let the expression be represented as $ x $, so:
\[
x = \sqrt{3 + \sqrt{3}}
\]
Now, square both sides to eliminate the square root:
\[
x^2 = 3 + \sqrt{3}
\]

### Step 2: Isolate the square root
Move the constant to the other side:
\[
x^2 - 3 = \sqrt{3}
\]

### Step 3: Square both sides again
To get rid of the square root, square both sides again:
\[
(x^2 - 3)^2 = (\sqrt{3})^2
\]
This simplifies to:
\[
(x^2 - 3)^2 = 3
\]
Expanding the left side:
\[
x^4 - 6x^2 + 9 = 3
\]
Simplify further:
\[
x^4 - 6x^2 + 6 = 0
\]

### Step 4: Solve the quartic equation
This is a quartic equation in $ x^2 $, so let $ y = x^2 $. The equation becomes:
\[
y^2 - 6y + 6 = 0
\]
Use the quadratic formula to solve for $ y $:
\[
y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(6)}}{2(1)}
\]
\[
y = \frac{6 \pm \sqrt{36 - 24}}{2}
\]
\[
y = \frac{6 \pm \sqrt{12}}{2}
\]
\[
y = \frac{6 \pm 2\sqrt{3}}{2}
\]
\[
y = 3 \pm \sqrt{3}
\]

### Step 5: Analyze the results
Since $ y = x^2 \geq 0 $, we take the positive root:
\[
x^2 = 3 + \sqrt{3}
\]
So, $ x = \sqrt{3 + \sqrt{3}} $, which is the original expression.

Therefore, the simplified form of the expression is:

\[
\boxed{\sqrt{3 + \sqrt{3}}}
\]

Would you like further details or clarifications? 

Here are 5 related questions to explore:
1. How do you simplify expressions involving nested square roots?
2. Can you derive general methods for solving quartic equations?
3. What are some properties of square roots that help in simplifications?
4. How does the quadratic formula apply to higher-degree polynomials?
5. What role do irrational numbers play in this solution?

**Tip:** Always verify by squaring your final answer to ensure it matches the original equation.

Learn how to simplify the nested square root expression $$ \sqrt{3 + \sqrt{3}} $$ through detailed algebraic steps. This solution involves solving a quartic equation and applying the quadratic formula. Suitable for students in Grades 9-12.

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