add and subtract radicals quick check

Answer 1

Here’s a quick guide to help you add and subtract radicals:

Adding and Subtracting Radicals

1. Like Radicals:

   • You can only add and subtract radicals that are "like" — that means they have the same radicand (the number under the radical).
   • Example:
     • \(\sqrt{5} + 3\sqrt{5} = 4\sqrt{5}\)
     • \(\sqrt{8} - \sqrt{2} \text{ cannot be combined as they are not like radicals.}\)

2. Unlike Radicals:

   • If the radicals are unlike (different radicands), you cannot combine them.
   • Example:
     • \(\sqrt{2} + \sqrt{3} \text{ remains as is.}\)

3. Finding Like Terms:

   • Simplify radicals first if possible.
   • Example:
     • \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\)
     • Then: \(3\sqrt{2} + \sqrt{2} = 4\sqrt{2}\)

Steps for Addition/Subtraction

1. Identify the radicals:

   • Are they like or unlike?

2. Simplify if needed:

   • Simplify any radicals to check if they can be combined.

3. Combine like terms:

   • Add or subtract the coefficients (the numbers in front) of the like radicals.

Example Problems

1. Example 1:

   • \(2\sqrt{3} + 4\sqrt{3} = (2 + 4)\sqrt{3} = 6\sqrt{3}\)

2. Example 2:

   • \(5\sqrt{7} - 3\sqrt{7} = (5 - 3)\sqrt{7} = 2\sqrt{7}\)

3. Example 3:

   • \(2\sqrt{5} + \sqrt{20}\)
   • First, simplify \(\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\)
   • Now combine: \(2\sqrt{5} + 2\sqrt{5} = 4\sqrt{5}\)

4. Example 4:

   • \(3\sqrt{2} - \sqrt{18} = 3\sqrt{2} - 3\sqrt{2} = 0\)

Practice Problems

1. \(4\sqrt{11} + 2\sqrt{11} = ?\)
2. \(5\sqrt{3} - 2\sqrt{3} = ?\)
3. \(\sqrt{12} + 3\sqrt{3} = ?\)
4. \(2\sqrt{5} + 5\sqrt{2} - \sqrt{5} = ?\)

Answers

1. \(6\sqrt{11}\)
2. \(3\sqrt{3}\)
3. \(4\sqrt{3}\)
4. \( (2\sqrt{5} - \sqrt{5}) + 5\sqrt{2} = \sqrt{5} + 5\sqrt{2}\)

This guide should help you quickly add and subtract radicals! Let me know if you have any questions or would like more practice problems.