Surds – Collecting Like Terms

Statement

When surds have the same base (the same number under the square root), they can be collected together just like algebraic terms.

\[ a\sqrt{b} + c\sqrt{b} = (a+c)\sqrt{b} \]

Why it’s true

• Think of \(\sqrt{b}\) as a “letter” (like x). For example, \(2\sqrt{3} + 5\sqrt{3}\) works the same as \(2x + 5x\).
• You cannot combine unlike surds (e.g. \(\sqrt{2} + \sqrt{3}\)).

Recipe (how to use it)

1. Check if the surds have the same base number inside the square root.
2. If they match, add or subtract the coefficients.
3. Keep the surd part the same.
4. If not the same, leave them separate (unless they can be simplified first).

Spotting it

Look for addition or subtraction with terms like \(k\sqrt{n}\). If the surds match, you can collect them; if not, simplify first to see if they become the same.

Common pairings

• Surd simplification before collecting.
• Expanding brackets with surds, then collecting like terms.

Mini examples

1. Given: \(3\sqrt{5} + 2\sqrt{5}\). Answer: \(5\sqrt{5}\).
2. Given: \(7\sqrt{2} - 4\sqrt{2}\). Answer: \(3\sqrt{2}\).

Pitfalls

• Trying to add different surds like \(\sqrt{2} + \sqrt{3}\).
• Forgetting to simplify surds first (e.g. \(\sqrt{50} = 5\sqrt{2}\)).

Exam strategy

• Always simplify surds before deciding if they are like terms.
• Check carefully: unlike surds cannot be combined.
• Treat surds as “algebra letters” for collecting terms.

Summary

Like surds can be combined by adding or subtracting their coefficients. Always check that the numbers under the square root are the same, and simplify first if needed.