Sergeant Loi’s Mid-Year Boot Camp 2008 – Rationalising The Rationale Of Surds

The Rationale of Surds

A. WHAT IS A SURD?

High-class definition:
An irrational root of a real number. *sweats*

Simple O Level definition:
A *square-root of a number that results in endless decimal places when you press your calculator e.g.

1. √2 is a surd (=1.4142135623 … when you press your calculator)
2. √9 is NOT a surd (= 3 when you press your calculator – WOW NO DECIMALS!)

*Actually can also be cube-root, n^th-root etc. but let’s not go there …

B. THE OPERATIONS OF SURDS

*Compare operations B(2) & B(3) to the Rules of Indices (Part C) and you’ll realize they’re like long-lost twins!

e.g. Simplifying surd expressions:

1. √18 = √(9 × 2) = √9 × √2 = 3√2
   → Using B(2) and a tiny bit of trial and error to get the (9 × 2)
2. 3√2 + 5√2 = (3 + 5)√2 = 8√2 → Using B(4)
3. (√5 – √2)² = (√5 – √2)(√5 – √2)
   = √5√5 – √5√2 – √2√5 + √2√2
   = 5 – √10 – √10 + 2 → Using B(1) + B(2)
   = 7 – 2√10 → Using B(4)

DON’T ever do this:

√(a+b) ≠ √a + √b → WRONG!
√(a–b) ≠ √a – √b → WRONG!

e.g.
√64 ≠ √32 + √32!
√25 ≠ √12 + √13!

C. CONJUGATE SURDS

e.g. (√3 + √2)(√3 – √2) = 3 – 2 = 1

DON’T ever do these:

(√a – √b)² ≠ (√a² – √b²) → WRONG!
(√a + √b)(√a – √b) ≠ (√a)² + (√b)² → WRONG!

D. RATIONALISING THE DENOMINATORS OF SURDS

Having a surd in the denominator is IMPROPER, UNGLAM, SINFUL & IMMORAL!!!

So we need to rationalise the denominators by multiplying the top and bottom by the same number with respect to the denominator i.e.

1. If denominator consists of a single term → multiply top & bottom by denominator term e.g.
   

2. If denominator consists of 2 terms → multiply top & bottom by conjugate of denominator e.g.