Mathematics • Year 10 • Unit 1 • Lesson 14

Surds and Operations, Skill Drill

Build fluency with the three PATHS surd skills from Lesson 14: simplify by extracting the largest perfect-square factor, combine like surds by adding coefficients, and rationalise a denominator by multiplying top and bottom by the surd.

Build · I Do / We Do / You Do

1. I do, fully worked example

Read every step. Each one has a short reason on the right so you can see why, not just what.

Problem. Simplify √72.

Step 1, Recall the simplest-form rule.

A surd is in simplest form when the number under the root has NO perfect-square factor (other than 1).

Reason: from the Simplifying Surds card. We always extract the largest perfect-square factor.

Step 2, Find the largest perfect-square factor of 72.

Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100, …

72 = 36 × 2, so the largest perfect square dividing 72 is 36.

Reason: 36 is the largest entry from our list that divides 72 cleanly.

Step 3, Apply the product rule.

√(a × b) = √a × √b

√72 = √(36 × 2) = √36 × √2

Reason: the product rule lets us split the radical across a multiplication.

Step 4, Evaluate the perfect square.

√36 = 6, so √72 = 6√2.

Answer: √72 = 6√2.

Stuck? Revisit lesson § "Simplifying Surds", Worked Example 1.

2. We do, fill in the missing steps

Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks

Problem. Rationalise the denominator of 3 / √5.

Step 1, Why rationalise? Mathematicians prefer denominators without surds. To remove √5 from the bottom, we multiply top AND bottom by __________.

Step 2, Set up the multiplication:

(3 / √5) × (______ / ______)

Step 3, Multiply the numerator.

3 × √5 = ______________

Step 4, Multiply the denominator. Recall √a × √a = a.

√5 × √5 = __________

Step 5, Write the final answer:

3 / √5 = __________ / __________

Stuck? Revisit lesson § "Rationalising the Denominator", Worked Example 3.

3. You do, independent practice

Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.

Foundation, single-step simplification

3.1 Simplify √48. 1 mark

3.2 Simplify √75. 1 mark

3.3 Simplify √200. 1 mark

3.4 Simplify √98. 1 mark

Standard, combine like surds and multiply

3.5 Simplify 3√2 + 5√2 − 2√2. 2 marks

3.6 Simplify √8 × √2. (Hint: use the product rule first, then simplify.) 2 marks

Extension, push your thinking

3.7 Simplify √50 + √18 − √8. (You will need to simplify each surd to make them like surds first.) 3 marks

3.8 Rationalise 2 / √3. 2 marks

Stuck on 3.7? Simplify each surd: √50 = 5√2, √18 = 3√2, √8 = 2√2. Now they are all like surds.

Answers, Do not peek before attempting

Section 2, We do (faded 3 / √5)

Step 1: multiply top and bottom by √5.
Step 2: (3 / √5) × (√5 / √5).
Step 3: 3 × √5 = 3√5.
Step 4: √5 × √5 = 5.
Step 5: 3 / √5 = 3√5 / 5.

3.1, √48

48 = 16 × 3. √48 = √16 × √3 = 4√3.

3.2, √75

75 = 25 × 3. √75 = √25 × √3 = 5√3.

3.3, √200

200 = 100 × 2. √200 = √100 × √2 = 10√2.

3.4, √98

98 = 49 × 2. √98 = √49 × √2 = 7√2.

3.5-3√2 + 5√2 − 2√2

All like surds (same √2). Combine coefficients: (3 + 5 − 2)√2 = 6√2.

3.6, √8 × √2

Product rule: √8 × √2 = √(8 × 2) = √16 = 4. (The surds collapse to a whole number because the product is a perfect square.)

3.7, √50 + √18 − √8

Simplify each: √50 = 5√2, √18 = 3√2, √8 = 2√2.
Now all like surds: 5√2 + 3√2 − 2√2 = (5 + 3 − 2)√2 = 6√2.

3.8, Rationalise 2 / √3

Multiply top and bottom by √3:
(2 / √3) × (√3 / √3) = 2√3 / (√3 × √3) = 2√3 / 3.