Indices and Surds
Senior 2 · Mathematics
Indices and Surds
Lesson 4 of 5
24 min
A square floor tile is 1 m long on each side. How long is its diagonal? By Pythagoras' theorem (Unit 6) the answer is √2 m. As a decimal, √2 = 1.414 213 56… and the digits never end or repeat. Any decimal you write is only an approximation.
Engineers and examiners keep such numbers in their exact surd form until the very last step, so that rounding errors do not build up. This lesson shows you how to simplify surds and calculate with them.
A surd is a root of a positive whole number (or fraction) that cannot be simplified to a whole number or fraction. Its decimal form never ends and never repeats, so it is irrational.
For a ≥ 0 and b ≥ 0 (and b ≠ 0 in the second rule):
Warning: there is no such rule for addition. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.
A surd is in its simplest form when the number under the root has no square factor other than 1. Split off the largest perfect square factor.
Worked example 1
Simplify √72.
The perfect squares are 4, 9, 16, 25, 36, 49, … The largest one that divides 72 is 36, and 72 = 36 × 2. So √72 = √36 × √2 = 6√2.
(If you use 4 first, √72 = 2√18 = 2 × 3√2 = 6√2. You reach the same answer, just in more steps.)
Only like surds, with the same number under the root, can be collected, just as 3x + 5x = 8x.
Multiply the whole numbers together and the surds together: 2√3 × 5√6 = 10√18 = 10 × 3√2 = 30√2.
Brackets are expanded exactly as in algebra.
Worked example 2
Expand and simplify (2 + √3)(4 − √3).
2 × 4 = 8, then 2 × (−√3) = −2√3, then √3 × 4 = 4√3, then √3 × (−√3) = −3. Total: 8 − 2√3 + 4√3 − 3 = 5 + 2√3.
Worked example 3
(√5 + √2)(√5 − √2) = 5 − √10 + √10 − 2 = 3. The middle terms cancel, which gives a rational answer. You will use this in the next lesson.
A square vegetable plot in Bugesera has an area of 50 m2. Find the exact length of its perimeter.
Side = √50 = √(25 × 2) = 5√2 m. Perimeter = 4 × 5√2 = 20√2 m. Only now use a calculator: 20√2 ≈ 28.28 m of fencing.
Tip
Memory hook. Learn the squares up to 152 = 225 by heart. When you simplify, search for the biggest square factor first (for √75 try 25 before 4 or 9). And remember that surds behave like algebra letters: you can add √3s to √3s, but not √3s to √2s.
Simplify √98.
Solution: 98 = 49 × 2, so √98 = 7√2.
Simplify √75 − √12.
Solution: √75 = 5√3 and √12 = 2√3. The difference is 3√3.
Simplify 3√2 × 4√10.
Solution: 3 × 4 = 12 and √2 × √10 = √20 = 2√5. So 12 × 2√5 = 24√5.
Expand (3 − √2)2.
Solution: (3 − √2)(3 − √2) = 9 − 3√2 − 3√2 + 2 = 11 − 6√2.
Which is larger, 3√5 or 2√11? Do not use a calculator.
Solution: Move each whole number inside the root: 3√5 = √(9 × 5) = √45 and 2√11 = √(4 × 11) = √44. Since 45 > 44, 3√5 is larger.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
