Sure! Let's go through each part step-by-step:
### Part (a)
Simplify [tex]\((5 + \sqrt{3})(3 + \sqrt{2})\)[/tex].
To simplify this, we use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(5 + \sqrt{3})(3 + \sqrt{2}) = 5 \cdot 3 + 5 \cdot \sqrt{2} + \sqrt{3} \cdot 3 + \sqrt{3} \cdot \sqrt{2}
\][/tex]
Simplifying each term individually:
[tex]\[
= 15 + 5\sqrt{2} + 3\sqrt{3} + \sqrt{6}
\][/tex]
So, the expression simplifies to:
[tex]\[
15 + 5\sqrt{2} + 3\sqrt{3} + \sqrt{6} \approx 29.7167
\][/tex]
### Part (b)
Simplify [tex]\(\sqrt{5} + \sqrt{3}\)[/tex].
This expression is already in its simplest form. It approximately equals:
[tex]\[
\sqrt{5} + \sqrt{3} \approx 3.9681
\][/tex]
### Part (c)
Simplify [tex]\((\sqrt{5} + \sqrt{3})^2\)[/tex].
Using the binomial expansion formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
[tex]\[
(\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2
\][/tex]
Simplifying each term individually:
[tex]\[
= 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}
\][/tex]
So, the expression simplifies to:
[tex]\[
8 + 2\sqrt{15} \approx 15.7460
\][/tex]
### Part (d)
Simplify [tex]\(\sqrt{5} - \sqrt{3}\)[/tex].
This expression is also already in its simplest form. It approximately equals:
[tex]\[
\sqrt{5} - \sqrt{3} \approx 0.5040
\][/tex]
Thus, summarizing the results:
- [tex]\((5 + \sqrt{3})(3 + \sqrt{2}) \approx 29.7167\)[/tex]
- [tex]\(\sqrt{5} + \sqrt{3} \approx 3.9681\)[/tex]
- [tex]\((\sqrt{5} + \sqrt{3})^2 \approx 15.7460\)[/tex]
- [tex]\(\sqrt{5} - \sqrt{3} \approx 0.5040\)[/tex]
