Certainly! Let's expand and simplify the expression [tex]\((\sqrt{3} + \sqrt{5})^2\)[/tex].
Step-by-step solution:
1. Understand the Expression:
[tex]\[
(\sqrt{3} + \sqrt{5})^2
\][/tex]
2. Apply the Square of a Binomial Formula:
The formula for the square of a binomial [tex]\((a + b)^2\)[/tex] is:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Here, [tex]\(a = \sqrt{3}\)[/tex] and [tex]\(b = \sqrt{5}\)[/tex].
3. Expand the Expression:
[tex]\[
(\sqrt{3} + \sqrt{5})^2 = (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{5}) + (\sqrt{5})^2
\][/tex]
4. Simplify Each Term:
- For the term [tex]\((\sqrt{3})^2\)[/tex]:
[tex]\[
(\sqrt{3})^2 = 3
\][/tex]
- For the term [tex]\(2(\sqrt{3})(\sqrt{5})\)[/tex]:
[tex]\[
2(\sqrt{3})(\sqrt{5}) = 2\sqrt{15}
\][/tex]
- For the term [tex]\((\sqrt{5})^2\)[/tex]:
[tex]\[
(\sqrt{5})^2 = 5
\][/tex]
5. Sum the Simplified Terms:
Adding together these simplified terms:
[tex]\[
3 + 2\sqrt{15} + 5
\][/tex]
6. Combine Like Terms:
Combine the constant terms:
[tex]\[
3 + 5 = 8
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
8 + 2\sqrt{15}
\][/tex]
7. Express the Result:
The expanded and simplified form of [tex]\((\sqrt{3} + \sqrt{5})^2\)[/tex] is:
[tex]\[
8 + 2\sqrt{15}
\][/tex]
Numerically, combining the values we have:
- [tex]\(8\)[/tex] (sum of the squared terms)
- [tex]\(2\sqrt{15}\)[/tex] approximately equals [tex]\(7.745966692414834\)[/tex]
Thus, if evaluated numerically:
[tex]\[
8 + 7.745966692414834 \approx 15.745966692414834
\][/tex]
So, the expanded and simplified expression is:
[tex]\[
8 + 2\sqrt{15} \approx 15.745966692414834
\][/tex]
Depending on the context, you can present the solution either in symbolic form, [tex]\(8 + 2\sqrt{15}\)[/tex], or its numerical approximation, [tex]\(15.745966692414834\)[/tex].
