Practice Maths

Simplifying Surds

Key Ideas

Key Terms

surd
An irrational root (e.g. √2, √5, √7).
Product rule
√(ab) = √a × √b  (a, b ≥ 0).
StepAction / Example
1. Find largest square factor√72: factors include 4, 9, 36. Largest = 36
2. Split√72 = √(36 × 2) = √36 × √2
3. Evaluate√36 × √2 = 6√2
4. With coefficient3√12 = 3 × 2√3 = 6√3
5. Add like surds6√3 + 5√3 − 2√3 = 9√3
Hot Tip Always use the largest perfect square factor. √72 = √(4×18) gives 2√18, which still needs simplifying. Using √(36×2) = 6√2 is done in one step.

Worked Example — Simplifying a Surd

Question: Simplify √72.

Step 1: Largest perfect square factor of 72 is 36.

Step 2: √72 = √(36 × 2) = √36 × √2 = 6√2.

Worked Example — Adding Like Surds

Question: Simplify 3√12 + √75 − 2√3.

3√12 = 6√3; √75 = 5√3.

6√3 + 5√3 − 2√3 = 9√3.

What Is a Surd?

A surd is a root that cannot be simplified to a rational number and is left in exact form using the √ symbol. For example, √2, √3, √7, and √50 (before simplifying) are all surds. √4 = 2 is not a surd because it simplifies to a rational number.

We prefer exact surd form over decimal approximations in many maths problems because it is precise. √2 is exact; 1.41421... is only an approximation.

Simplifying Surds: Finding the Largest Perfect Square Factor

To simplify a surd like √72, rewrite the number under the root as a product where one factor is the largest possible perfect square. The perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

For √72: the largest perfect square factor is 36. So: √72 = √(36 × 2) = √36 × √2 = 6√2.

This uses the product rule: √(a × b) = √a × √b. Always pick the largest perfect square factor so you only need to simplify once.

More examples: √50 = √(25 × 2) = 5√2. √48 = √(16 × 3) = 4√3. √75 = √(25 × 3) = 5√3.

The Quotient Rule for Surds

The quotient rule states: √a / √b = √(a/b). This is useful for simplifying expressions like √(75/3) = √25 = 5, or when dividing surds: √24 / √6 = √(24/6) = √4 = 2.

Adding and Subtracting Like Surds

You can only add or subtract surds that have the same number under the root sign — these are called like surds. It works exactly like collecting like terms in algebra.

3√2 + 5√2 = 8√2 (just as 3x + 5x = 8x). But √3 + √5 cannot be simplified — they are unlike surds.

Sometimes you need to simplify first before you can add: √12 + √27 = 2√3 + 3√3 = 5√3. Always simplify before trying to add or subtract.

Exam strategy: Always check you have found the largest perfect square factor. A common mistake is using a smaller factor: √72 = √(4 × 18) = 2√18, but √18 can be simplified further to 3√2, giving 6√2 in the end. Using the largest factor gets you there in one step. Ask yourself: can the number under my remaining root be simplified further? If yes, you have not finished.

Mastery Practice

  1. Simplify each surd fully. Show which perfect square factor you used. Fluency

      Surd Simplified Form
    (a)√8 
    (b)√18 
    (c)√50 
    (d)√75 
    (e)√12 
    (f)√45 
    (g)√72 
    (h)√98 

  2. Simplify each expression involving a coefficient. Fluency

      Expression Simplified Form
    (a)2√12 
    (b)3√20 
    (c)5√8 
    (d)4√27 
    (e)2√45 
    (f)3√50 
    (g)½√48 
    (h)6√75 

  3. Expand each expression and simplify. Fluency

      Expression Expanded & Simplified
    (a)√2(3 + √2) 
    (b)√3(2√3 − √12) 
    (c)2√5(3√5 + √20) 
    (d)(√6 + 2)(√6 − 2) 

  4. State whether each claim is True or False, with justification. Fluency

    1. √20 = 2√5
    2. √8 + √2 = √10
    3. 3√2 = √18
    4. √(a + b) = √a + √b for all positive a, b
    5. (√5 + √5)² = 20
    6. √200 = 10√2

  5. Rectangle dimensions in surd form. Understanding

    Framing a Photo. A rectangular photo frame has length √50 cm and width √8 cm.
    1. Find the exact perimeter in simplest surd form.
    2. Find the exact area. Explain why the area is a whole number.
    3. A second frame has length 3√2 cm and width 2√2 cm. Find its perimeter and area, and compare both dimensions to the first frame.

  6. Equilateral triangle with surd side. Understanding

    Triangle Design. An equilateral triangle has side length √12 cm.
    1. Find the exact perimeter in simplest surd form.
    2. Use the formula Area = (√3/4)s² to find the exact area. Simplify fully.
    3. A student says the perimeter is “about 10.4 cm.” Verify this using a calculator.

  7. Square with known area. Understanding

    Square Patch. A square patch of grass has an area of 48 m².
    1. Find the exact side length in simplest surd form.
    2. Find the exact perimeter in simplest surd form.
    3. Is the perimeter rational or irrational? Explain.

  8. Compare two squares using surd arithmetic. Understanding

    Two Garden Squares. Square A has area 18 m². Square B has area 50 m².
      Square A (area 18 m²) Square B (area 50 m²)
    Side length  
    Perimeter  

    Show without a calculator that the side of Square B is exactly 5/3 times the side of Square A.

  9. Pythagoras’ theorem and surd simplification. Problem Solving

    Right-Angle Problems. Use c² = a² + b² and simplify all answers fully.
    1. A right triangle has legs √5 cm and √20 cm. Find the exact hypotenuse.
    2. A right triangle has legs 2√3 cm and √3 cm. Find the exact hypotenuse.
    3. A diagonal of a rectangle is √108 cm. One side is 6 cm. Find the other side in simplest surd form.

  10. Surd spiral investigation. Problem Solving

    Surd Spiral. A spiral is constructed by placing right triangles end-to-end. The first triangle has legs 1 and 1, giving hypotenuse √2. The next has legs √2 and 1, giving hypotenuse √3, and so on.
    1. Write down the hypotenuse of the 4th, 5th, and 6th triangles in this sequence.
    2. Find the simplified surd form for √8 and √12, which appear as hypotenuses in the sequence.
    3. Which hypotenuse in the sequence is the first to be rational? Justify your answer.
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