Certainly! Let's expand and simplify the given expression step by step:
We need to simplify the expression [tex]\((\sqrt{6} + \sqrt{3})(\sqrt{6} - \sqrt{3})\)[/tex].
1. Recognize the Structure:
The expression [tex]\((\sqrt{6} + \sqrt{3})(\sqrt{6} - \sqrt{3})\)[/tex] is in the form of a difference of squares, which can be represented as:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
2. Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
Here, [tex]\(a = \sqrt{6}\)[/tex] and [tex]\(b = \sqrt{3}\)[/tex].
3. Apply the Formula:
We apply the difference of squares formula:
[tex]\[
(\sqrt{6} + \sqrt{3})(\sqrt{6} - \sqrt{3}) = (\sqrt{6})^2 - (\sqrt{3})^2
\][/tex]
4. Calculate Each Term:
- Calculate [tex]\((\sqrt{6})^2\)[/tex]:
[tex]\[
(\sqrt{6})^2 = 6
\][/tex]
- Calculate [tex]\((\sqrt{3})^2\)[/tex]:
[tex]\[
(\sqrt{3})^2 = 3
\][/tex]
5. Subtract the Results:
Now, subtract the second term from the first:
[tex]\[
6 - 3 = 3
\][/tex]
6. Conclusion:
Therefore, the simplified form of [tex]\((\sqrt{6} + \sqrt{3})(\sqrt{6} - \sqrt{3})\)[/tex] is:
[tex]\[
3
\][/tex]
So, [tex]\((\sqrt{6} + \sqrt{3})(\sqrt{6} - \sqrt{3}) = 3\)[/tex].
