Practice Maths

Simplifying Surds

Key Terms

A surd is an irrational number expressed using a radical sign (√). It cannot be written as an exact fraction.
√4 = 2 is not a surd (it simplifies to a rational number); √5 is a surd.
To simplify a surd, find the largest perfect square factor of the radicand: √(a × b) = √a × √b.
Like surds
have the same radical part and can be added or subtracted like algebraic like terms.
When multiplying surds: √a × √b = √(ab); also m√a × n√b = mn√(ab).
(√a)² = a for a ≥ 0.
OperationRule / Example
Simplify√45 = √(9×5) = 3√5
Add like surds3√2 + 5√2 = 8√2
Subtract like surds7√3 − 2√3 = 5√3
Multiply surds2√3 × 5√11 = 10√33
Square a surd(√7)² = 7
Unlike surds√2 + √3 — cannot be simplified further
Hot Tip Always find the largest perfect square factor first. Simplifying √72 as √(4×18) = 2√18 is not fully simplified — you then need to simplify √18 again. Instead, spot that 36 is a factor: √(36×2) = 6√2 in one step.

Worked Example 1 — Simplify and combine surds

Question: Simplify √50 − 2√8 + √18.

Step 1: Simplify each surd: √50 = √(25×2) = 5√2; √8 = √(4×2) = 2√2; √18 = √(9×2) = 3√2.

Step 2: Substitute back: 5√2 − 2(2√2) + 3√2 = 5√2 − 4√2 + 3√2.

Step 3: Collect like surds: (5 − 4 + 3)√2 = 4√2.

Answer: 4√2

Worked Example 2 — Multiply surds

Question: Simplify 3√6 × 2√15.

Step 1: Multiply the coefficients: 3 × 2 = 6.

Step 2: Multiply the radicands: √6 × √15 = √90.

Step 3: Simplify √90: √(9×10) = 3√10.

Step 4: Combine: 6 × 3√10 = 18√10.

Answer: 18√10

What IS a Surd?

A surd is a root that cannot be simplified to a rational number. When we write √5, we are not being lazy — we are being exact. The decimal 2.2360679… goes on forever without repeating, so no fraction can capture it precisely. √5 is the only way to write the exact value.

Contrast this with √9: since 9 = 3², we get √9 = 3, a rational number, so √9 is not a surd. The test is simple: does the square root simplify to a terminating or repeating decimal? If yes, not a surd. If no, it is a surd.

In higher mathematics and physics, exact answers are essential. A calculator gives √2 ≈ 1.41421356… but that is already wrong — it is a rounded approximation. Whenever a problem says “exact value”, surd form is the answer.

Surds as Lengths: Where They Really Come From

Surds are not invented for convenience — they arise naturally from geometry. Draw a right triangle with both legs of length 1. By Pythagoras’ theorem, the hypotenuse has length √(1² + 1²) = √2. That length is real and physical. You can draw it. You just cannot write it as a fraction.

Similarly, an equilateral triangle with side length 2 has height √3 (found by splitting it into two 30–60–90 triangles). These lengths give rise to the exact trig values you will use throughout Year 11 and 12. Understanding surds geometrically makes them far less abstract.

Multiplying and Dividing Surds: Why the Rules Work

The product rule √a × √b = √(ab) is not just a rule to memorise — it follows from the definition of a square root. If we square both sides of the equation √a × √b = √(ab), we get (√a)² × (√b)² = a × b on the left, and ab on the right. Both sides equal ab, so the equation is true (for a, b ≥ 0).

This rule lets us split a radicand: √12 = √(4 × 3) = √4 × √3 = 2√3. The reverse direction is equally important: 3√6 × 2√5 = 6√30, combining coefficients and radicands separately.

Division works similarly: √a / √b = √(a/b). This underlies rationalising denominators in the next lesson.

Simplifying by Finding Square Factors

To simplify √n, find the largest perfect square that divides n. Perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

For √72: the factors of 72 include 4, 9, and 36. The largest perfect square factor is 36, so √72 = √(36 × 2) = 6√2. If you only spotted 4, you would get √72 = 2√18 — correct but not fully simplified (since √18 = 3√2 still needs simplifying). Always aim for the largest perfect square factor in one step.

A systematic approach: write the prime factorisation. 72 = 2³ × 3². Group prime pairs: 2² × 3² = 36 is the perfect square part, leaving one factor of 2. So √72 = √(36 × 2) = 6√2.

Adding and Subtracting Surds: Like Terms Only

Adding surds works exactly like adding algebraic like terms. Just as 3x + 5x = 8x (you add coefficients, keep the variable), we have 3√2 + 5√2 = 8√2. The √2 part is the “type” of surd — you can only add surds of the same type.

You cannot simplify √2 + √3 further, just as you cannot add 3x + 5y into a single term. The expressions represent different irrational numbers (approximately 1.414 and 1.732), and their sum is approximately 3.146 — but there is no cleaner surd form.

The hidden skill is simplifying before adding. √8 + √18 looks like unlike surds, but √8 = 2√2 and √18 = 3√2, so √8 + √18 = 2√2 + 3√2 = 5√2. Always simplify each surd fully before attempting to collect like terms.

Exam Tip: A very common error is to “add under the root”: students write √9 + √16 = √25 = 5. This is wrong. √9 + √16 = 3 + 4 = 7, not 5. Square roots do not distribute over addition. In general, √(a + b) ≠ √a + √b. If you are ever unsure, substitute numbers to check.
Exam Tip: Always check your final answer has no perfect square factors remaining inside the surd. If you get √50 as an answer, stop and simplify: √50 = 5√2. Examiners expect fully simplified surds. A quick check: if the number inside is divisible by 4, 9, 25, 49 … it is not fully simplified.

Mastery Practice

See Answers ➔
  1. Fluency

    State whether each of the following is a surd or not a surd. Justify your answer.

    1. (a) √16
    2. (b) √7
    3. (c) √0.25
    4. (d) √11
    5. (e) √(4/9)
    6. (f) √50
  2. Fluency

    Simplify each surd by extracting the largest perfect square factor.

    1. (a) √12
    2. (b) √45
    3. (c) √75
    4. (d) √98
    5. (e) √200
    6. (f) √147
  3. Fluency

    Simplify each expression by collecting like surds.

    1. (a) 3√5 + 7√5
    2. (b) 9√3 − 4√3
    3. (c) 2√7 + 5√7 − √7
    4. (d) 4√11 − 6√11 + 3√11
    5. (e) 5√2 + √2 − 3√2
  4. Fluency

    Simplify each expression. First simplify the surds, then collect like terms.

    1. (a) √8 + √18
    2. (b) √12 + √27
    3. (c) √50 − √8
    4. (d) 2√45 + √20
    5. (e) √75 − √48 + √3
  5. Fluency

    Simplify each product.

    1. (a) √5 × √5
    2. (b) √3 × √12
    3. (c) 2√7 × 3√7
    4. (d) 4√3 × 2√5
    5. (e) 3√2 × √8
    6. (f) 5√6 × 2√6
  6. Understanding

    Mixed simplification.

    Combining skills. These expressions require you to simplify surds and then add or subtract. Some results may not simplify to like surds.
    1. (a) √50 − 2√8 + √18
    2. (b) 3√12 + √75 − 2√27
    3. (c) √98 + √50 − √72
    4. (d) 2√80 − √45 + 3√20
  7. Understanding

    Expanding with surds.

    Distributive law. Expand and simplify each expression using the distributive law.
    1. (a) √3(2 + √3)
    2. (b) √2(3√2 − √8)
    3. (c) 2√5(3 + √5)
    4. (d) √6(2√6 − √3 + 1)
  8. Understanding

    Perimeter and area with surds.

    Geometry context. A rectangle has length 3√5 cm and width √20 cm.
    1. (a) Simplify the width √20 cm.
    2. (b) Find the perimeter of the rectangle in simplified surd form.
    3. (c) Find the area of the rectangle. Express your answer as an integer.
  9. Problem Solving

    Diagonal of a rectangle.

    Challenge. A rectangular room measures √48 m by √75 m. Using Pythagoras' theorem, find the exact length of the diagonal of the room in fully simplified surd form. Then calculate the perimeter and explain why the perimeter can be expressed as an integer multiple of √3.
    1. (a) Simplify the dimensions of the room.
    2. (b) Use Pythagoras' theorem to find the exact length of the diagonal.
    3. (c) Find the perimeter in simplified form and express it as k√3, stating the value of k.
  10. Problem Solving

    Proof and surds.

    Challenge. Consider the expression (√a + √b)² where a and b are positive integers.
    1. (a) Expand (√a + √b)² and simplify.
    2. (b) Use your result to find the exact value of (√3 + √12)² without using a calculator.
    3. (c) Show that (√5 + √20)² = 45. Explain what this tells you about √5 + √20.