We are tasked with simplifying the expression:

\[
\sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}
\]

Let's try to simplify this step by step.

### Step 1: Assign variables to the square roots
Let:
\[
x = \sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}
\]

### Step 2: Square both sides to eliminate the square roots
Squaring both sides of the equation:
\[
x^2 = \left( \sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}} \right)^2
\]

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we get:
\[
x^2 = \left( \sqrt{3 + \sqrt{5}} \right)^2 + 2\sqrt{(3 + \sqrt{5})(3 - \sqrt{5})} + \left( \sqrt{3 - \sqrt{5}} \right)^2
\]

### Step 3: Simplify each term
1. \(\left( \sqrt{3 + \sqrt{5}} \right)^2 = 3 + \sqrt{5}\)
2. \(\left( \sqrt{3 - \sqrt{5}} \right)^2 = 3 - \sqrt{5}\)
3. Now, simplify the middle term:
   \[
   \sqrt{(3 + \sqrt{5})(3 - \sqrt{5})} = \sqrt{9 - (\sqrt{5})^2} = \sqrt{9 - 5} = \sqrt{4} = 2
   \]

Thus, we have:
\[
x^2 = (3 + \sqrt{5}) + (3 - \sqrt{5}) + 2(2)
\]

### Step 4: Combine like terms
Simplifying the expression:
\[
x^2 = 3 + \sqrt{5} + 3 - \sqrt{5} + 4
\]
\[
x^2 = 6 + 4 = 10
\]

### Step 5: Take the square root of both sides
\[
x = \sqrt{10}
\]

Thus, the simplified form of the expression is:
\[
\boxed{\sqrt{10}}
\]

Would you like more details on any step? Here are some related questions to deepen your understanding:

1. How can you simplify similar expressions with square roots?
2. Why do we square both sides to eliminate the square roots?
3. Can this technique be used for complex numbers?
4. What other methods can simplify nested square roots?
5. How does the distributive property work in squaring binomials?

**Tip:** When dealing with square roots, always try simplifying under the square root first and then apply algebraic techniques like squaring both sides.

Discover how to simplify the expression √(3 + √5) + √(3 − √5) through step-by-step algebraic techniques. This problem highlights the use of the distributive property and the properties of radicals, perfect for students in Grades 9-12.

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