Adding and subtracting surds
Surds often pop up when we deal with square roots that can’t be simplified to whole numbers – like in measurements or areas involving diagonals. Learning to add and subtract them is a bit like collecting like terms in algebra: once you spot the matching roots, you can tidy them up neatly! Jump to the questions
Practise now
Enter all answers in their simplest form.
Topic guide
What this worksheet practises
This worksheet provides practice on adding and subtracting surds. Just like in algebra where you can only add "like terms" (such as 2x + 3x), you can only add or subtract surds if the number inside the square root is exactly the same.
Key method
To add or subtract surds, you must first simplify them to see if they share a common base surd.
- First, simplify each surd in the expression. Look for the largest square number that divides into the number under the root.
- Second, rewrite the surd as a product of the square number and the remaining factor, and take the square root of the square number outside the radical.
- Finally, collect any "like surds" together by adding or subtracting the numbers outside the square roots, treating the surd itself like a letter in algebra.
Worked example
Simplify √12 + √27
Step 1: Simplify √12. The largest square number factor is 4.
√12 = √(4 × 3) = √4 × √3 = 2√3.
Step 2: Simplify √27. The largest square number factor is 9.
√27 = √(9 × 3) = √9 × √3 = 3√3.
Step 3: Add the simplified like surds together.
2√3 + 3√3 = 5√3.
Common mistakes to avoid
The most common mistake is attempting to add the numbers inside the square roots directly. For example, incorrectly assuming that √2 + √3 equals √5. You can only ever add the coefficients outside the surds, and only when the numbers inside the square roots are identical.
Things to remember
The first 10 square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Memorising these makes finding the largest square factor of a surd much faster.