https://r2.erweima.ai/mathbot/image/1726068809385adza4ng5.jpg

The given mathematical expression in the image is:

\[
\sqrt{2 - \sqrt{3}} + \sqrt{2 + \sqrt{3}} = \, ?
\]

To simplify this, let's follow these steps:

### Step 1: Use substitution.
Let’s assume:
\[
a = \sqrt{2 - \sqrt{3}} \quad \text{and} \quad b = \sqrt{2 + \sqrt{3}}
\]
Thus, the expression becomes:
\[
a + b
\]

### Step 2: Square both expressions.
Now, let's square each part individually.

\[
a^2 = 2 - \sqrt{3}
\]
\[
b^2 = 2 + \sqrt{3}
\]

### Step 3: Add the squared expressions.
\[
a^2 + b^2 = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 4
\]
This gives us:
\[
a^2 + b^2 = 4
\]

### Step 4: Multiply both expressions.
Next, let's calculate the product \(ab\):
\[
ab = \sqrt{(2 - \sqrt{3})(2 + \sqrt{3})}
\]
Using the difference of squares:
\[
ab = \sqrt{(2^2 - (\sqrt{3})^2)} = \sqrt{4 - 3} = \sqrt{1} = 1
\]

### Step 5: Use the identity.
Now, apply the identity for the square of a sum:
\[
(a + b)^2 = a^2 + b^2 + 2ab
\]
Substitute the known values:
\[
(a + b)^2 = 4 + 2(1) = 6
\]
Thus:
\[
a + b = \sqrt{6}
\]

### Final Answer:
\[
\sqrt{2 - \sqrt{3}} + \sqrt{2 + \sqrt{3}} = \sqrt{6}
\]

Would you like further explanation or details on any step? Here are some related questions to explore further:

1. How is the difference of squares used in simplifying the expression?
2. Why is it helpful to square both sides when dealing with square roots?
3. Can this method be applied to other expressions involving nested square roots?
4. What are other possible ways to simplify such radicals?
5. How would this expression change if the signs within the square roots were different?

**Tip:** When dealing with square roots, squaring both sides is often a helpful approach to eliminate radicals and simplify the expression.

Explore how to simplify the expression √(2 - √3) + √(2 + √3) using radical properties and the difference of squares theorem. A step-by-step solution demonstrates key algebraic concepts, making this suitable for students in Grades 9-12.

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