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The expression provided is:

\[
\sqrt{8 + 3\sqrt{3}} \cdot \sqrt{8 - 3\sqrt{3}}
\]

To simplify this expression, let's use the identity:

\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\]

So, the given expression can be rewritten as:

\[
\sqrt{(8 + 3\sqrt{3})(8 - 3\sqrt{3})}
\]

Now, notice that this is a difference of squares, which has the form:

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

In this case, \(a = 8\) and \(b = 3\sqrt{3}\), so:

\[
(8 + 3\sqrt{3})(8 - 3\sqrt{3}) = 8^2 - (3\sqrt{3})^2
\]

Calculating each term:

\[
8^2 = 64
\]

\[
(3\sqrt{3})^2 = 9 \times 3 = 27
\]

Thus:

\[
64 - 27 = 37
\]

So the expression simplifies to:

\[
\sqrt{37}
\]

Therefore, the simplified form of the given expression is \(\sqrt{37}\).

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How would you simplify \(\sqrt{a + b} \times \sqrt{a - b}\) for different values of \(a\) and \(b\)?
2. Can you apply the difference of squares identity to other algebraic expressions?
3. What are the steps to simplify a product of square roots without using the difference of squares?
4. How would the process differ if one of the terms inside the square root involved a negative square root?
5. What is the significance of recognizing patterns like the difference of squares in algebra?

**Tip:** When simplifying square roots, look for opportunities to apply algebraic identities, such as the difference of squares, to make the calculation easier.

Simplify the expression: \(\sqrt{8 + 3\sqrt{3}} \cdot \sqrt{8 - 3\sqrt{3}}\)

Discover how to simplify the expression \(\sqrt{8 + 3\sqrt{3}} \cdot \sqrt{8 - 3\sqrt{3}}\) using the difference of squares identity. This detailed solution explores key algebraic concepts and is perfect for students in Grades 8-10.

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