The surd addition and subtraction rules are based on the idea that you can only combine surds (square roots, cube roots, etc.) that have the same radicand (the number under the root). Here’s how the rules work:

Addition and Subtraction of Surds

• Same radicand: You can add or subtract surds only when they have the same radicand (the same number under the root symbol).
• Simplifying the surd: If the surd has a common factor, you can factor it out first before adding or subtracting.

Rule for Addition/Subtraction

• If you have two surds with the same radicand, say (\sqrt{a}) and (\sqrt{a}), then: $$\sqrt{a} + \sqrt{a} = 2\sqrt{a}$$ $$\sqrt{a} - \sqrt{a} = 0$$ Essentially, you combine the coefficients (the numbers in front of the surd) and keep the same radicand.

Examples

1. Addition:

   $$3\sqrt{2} + 5\sqrt{2} = (3 + 5)\sqrt{2} = 8\sqrt{2}$$

2. Subtraction:

   $$6\sqrt{5} - 2\sqrt{5} = (6 - 2)\sqrt{5} = 4\sqrt{5}$$

Rule for Different Radicands

If the radicands are different, you cannot directly add or subtract them. For example:

$$\sqrt{2} + \sqrt{3} \neq \sqrt{5}$$

In this case, you would leave the terms separate.

Simplifying Before Addition/Subtraction

If the surds can be simplified, simplify them first. For example:

$$\sqrt{18} + \sqrt{8}$$

Simplify (\sqrt{18}) and (\sqrt{8}):

$$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$

Now add the terms:

$$3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$$

In summary:

• Same radicand: Add/subtract the coefficients, keeping the radicand the same.
• Different radicands: Leave the terms separate.
• Simplify first: If possible, simplify the surds before performing operations.

Latex

a \sqrt{b} + c \sqrt{b} = (a + c) \sqrt{b}
a \sqrt{b} - c \sqrt{b} = (a - c) \sqrt{b}