Adding and subtracting surds

Question 1

What is a surd?

Answer 1

A surd is a square root (or other root) that cannot be simplified to a whole number.

Examples: √2, √3, √5 = surds (irrational); √4 = 2 (not a surd — it’s rational!)

Question 2

When can you add or subtract surds?

Answer 2

Only when the radical part is the same — like terms only!

✅ √3 + 2√3 = 3√3; ❌ √3 + √2 = leave it like that, they aren’t compatible

Question 3

How do you simplify surds?

Answer 3

Break the surd into square factors:

Example: √50 = √(25 × 2) = √25 × √2 = 5√2

Question 4

Why simplify surds before adding or subtracting?

Answer 4

So you can see if they’re like terms!

Example: √50 + √8 👉 Simplify first: 5√2 + 2√2 = 7√2

Question 5

Simplify: √18 + √8

Answer 5

Break ‘em down:

√18 = √(9×2) = 3√2
√8 = √(4×2) = 2√2
Add like terms: 3√2 + 2√2 = 5√2

Question 6

What do I do if surds aren’t like terms?

Answer 6

If they can’t be simplified to the same radical, leave them.

Example: √3 + √7 = can’t simplify, so keep it like that.

Question 7

What are common mistakes when adding/subtracting surds?

Answer 7

❌ Trying to add unlike surds (e.g. √5 + √7 ≠ √12)
❌ Adding the numbers inside the root (e.g. √2 + √2 ≠ √4 — it’s 2√2)
✅ Always simplify before adding/subtracting!