What Is a Surd?
A surd is the square root of a non-perfect square number. The word "surd" refers to an irrational number—a number that cannot be written as a simple fraction and whose decimal expansion goes on forever without repeating. Using surds allows you to express answers exactly, without rounding or approximation.
Surd: The square root of a positive integer that is not a perfect square. Examples: √2, √7, √13, √23. These cannot be simplified to a rational number.
Surds vs. Non-Surds
These ARE surds: √2, √3, √5, √7, √11 — because 2, 3, 5, 7, and 11 are not perfect squares.
These are NOT surds: √4 = 2, √9 = 3, √16 = 4 — because the results are whole numbers (rational). Also, 2/3 and 6.23 are not surds because they aren't square roots of integers.
Why use surds? The exact value √5 is more accurate and elegant than the decimal approximation 2.236... Using surds preserves precision throughout your working and avoids accumulated rounding errors.
Operations with Surds
Surds follow special rules for multiplication, division, addition, and subtraction. Understanding these rules is essential for working with surds confidently.
Multiplying Surds
You can multiply surds by multiplying the numbers under the square roots together, then simplifying.
Multiplying Surds
√a × √b = √(ab)
Example: √3 × √5
√3 × √5 = √(3 × 5) = √15
Dividing Surds
You can divide surds by dividing the numbers under the square roots, then simplifying.
Dividing Surds
√a ÷ √b = √(a ÷ b) = √(a/b)
Example: √21 ÷ √7
√21 ÷ √7 = √(21 ÷ 7) = √3
Adding and Subtracting Surds
You can only add or subtract surds if they are "like" surds—meaning the same number is under the square root.
Like Surds (Can Be Added/Subtracted)
3√5 + 8√5 = 11√5 (same surd: √5)
7√3 − 4√3 = 3√3 (same surd: √3)
Unlike Surds (Cannot Be Added/Subtracted)
2√3 + 4√6 — cannot be simplified (different surds)
√9 + √4 = 3 + 2 = 5 (these are not surds; they simplify to whole numbers)
⚠ Critical Mistake
√9 + √4 ≠ √(9 + 4). Never add numbers inside the square root! √9 + √4 = 3 + 2 = 5, but √13 ≈ 3.6.
How Do I Simplify Surds?
To simplify a surd, you extract any perfect square factors from under the square root. This is the most important skill for working with surds.
Simplifying Surds Strategy
√(a × b) = √a × √b
Find the largest perfect square that divides the number, then separate it out.
Step-by-Step Process
- Find the largest perfect square factor of the number under the root