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.

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What I'll revisit before next class:

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.