Sure! Let's create two problems involving all four operations (addition, subtraction, multiplication, and division) of radicals.
### Problem 1: Addition and Multiplication
Let's solve the following problem:
\[
\sqrt{8} + 3 \cdot \sqrt{2} + \sqrt{18}
\]
We can simplify the radicals first:
- \(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\)
- \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\)
Substituting back into the equation:
\[
2\sqrt{2} + 3\sqrt{2} + 3\sqrt{2}
\]
Now, combine the like terms:
\[
(2 + 3 + 3)\sqrt{2} = 8\sqrt{2}
\]
So, the solution to Problem 1 is:
\[
\sqrt{8} + 3 \cdot \sqrt{2} + \sqrt{18} = 8\sqrt{2}
\]
### Problem 2: Subtraction and Division
Now let's tackle the following problem:
\[
\sqrt{50} - \frac{\sqrt{18}}{\sqrt{2}}
\]
We will simplify each term.
1. Simplifying \(\sqrt{50}\):
\[
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
\]
2. Simplifying \(\frac{\sqrt{18}}{\sqrt{2}}\):
\[
\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3
\]
Now substituting these back into the expression:
\[
5\sqrt{2} - 3
\]
This is as simplified as it gets. Therefore, the solution to Problem 2 is:
\[
\sqrt{50} - \frac{\sqrt{18}}{\sqrt{2}} = 5\sqrt{2} - 3
\]
### Summary of Solutions:
- **Problem 1**: \(\sqrt{8} + 3 \cdot \sqrt{2} + \sqrt{18} = 8\sqrt{2}\)
- **Problem 2**: \(\sqrt{50} - \frac{\sqrt{18}}{\sqrt{2}} = 5\sqrt{2} - 3\)
These problems demonstrate the addition, multiplication, subtraction, and division of radicals while simplifying correctly.