Express the product left parenthesis, 3, minus, square root of, 5, end square root, right parenthesis, left parenthesis, 3, plus, square root of, 5, end square root, right parenthesis(3− 
5
​
 )(3+ 
5
​
 ) in simplest form.

The given expression is \((3 - \sqrt{5})(3 + \sqrt{5})\).

This is a difference of squares, which follows the formula:

\[
(a - b)(a + b) = a^2 - b^2
\]

Here, \( a = 3 \) and \( b = \sqrt{5} \). Applying the formula:

\[
(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2
\]

Now, simplify:

\[
3^2 = 9 \quad \text{and} \quad (\sqrt{5})^2 = 5
\]

Thus:

\[
9 - 5 = 4
\]

So, the simplest form of the expression is:

\[
\boxed{4}
\]

Would you like more details or further explanation?

Here are some related questions:
1. How does the difference of squares formula work?
2. What is the general process for simplifying radical expressions?
3. Can we apply the difference of squares for other complex numbers?
4. How do we simplify expressions involving square roots?
5. What are some examples where the difference of squares appears in geometry?

**Tip:** Remember, the difference of squares formula is useful for simplifying expressions with conjugates like \((a - b)(a + b)\).

Express the product (3 - √5)(3 + √5) in simplest form.

Learn how to simplify the expression (3 - √5)(3 + √5) using the difference of squares formula. This algebraic concept simplifies expressions involving radicals and is suitable for students in Grades 8-10.

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