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.
-
Simplifying \(\sqrt{50}\): \[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \]
-
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.