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2008
Sun
4
May
Sergeant Loi avatar

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

(7)
Tuition given in the topic of A-Maths Tuition Notes & Tips from the desk of Sergeant Loi at 12:38 am (Singapore time)
Surds

Don’t worry.
These aliens are often friendlier than they look.

*Puts on helmet*

*SHOWS STERN & MEAN FACE*

*Sounds bugle* EVERYBODY WAKE UP!

Sergeant Loi is back from her May Day leave – only to find The Temple in a state of 兵慌马乱 as the mid-year exams loom.

To whip everyone back to shape, Sergeant Loi shall complete the trinity (which began with indices, and followed by logarithms) with the concluding drill on surds today!

Despite the scary-looking longish workings, surds are the most straightforward part of the trinity. For most parts, the TWO main things you’ll ever need to know are how to simplify (with a dose of trial-and-error common sense) and rationalise surd expressions.

That’s it! Now let’s go eat some surds for breakfast!

*Cracks whip*

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, nth-root etc. but let’s not go there …

B. THE OPERATIONS OF SURDS

  1. sqrt{a}*sqrt{a} = a
  2. sqrt{a}*sqrt{b} = sqrt{ab}
  3. sqrt{a}/sqrt{b} = sqrt{a/b}
  4. m sqrt{a}+n sqrt{a} = (m+n)sqrt{a}
  5. m sqrt{a}-n sqrt{a} = (m-n)sqrt{a}

*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)
Warning! DON’T ever do this:

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

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

C. CONJUGATE SURDS

(ha + kb)(hakb) = h2a k2b
(Note the MINUS sign on the RHS)

e.g. (√3 + √2)(√3 – √2) = 3 – 2 = 1

Warning! 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.
    1/sqrt{3} = {1/sqrt{3}} * {sqrt{3}/sqrt{3}} = sqrt{3}/3
  2. If denominator consists of 2 terms → multiply top & bottom by conjugate of denominator e.g.
    1/{sqrt{2}-1} = {1/{sqrt{2}-1}} * {{sqrt{2}+1}/{sqrt{2}+1}} = {sqrt{2}+1}/{(sqrt{2}-1)(sqrt{2}+1)}
    {}= {sqrt{2}+1}/{sqrt{2}^2-1^2} = {sqrt{2}+1}/{2-1} = sqrt{2}+1

SAMPLE PRACTICE QUESTIONS

  1. Simplify {4-sqrt{5}}/{2sqrt{5}+3}.

    Ans:

    {{4-sqrt{5}}/{2sqrt{5}+3}} * {{2sqrt{5}-3}/{2sqrt{5}-3}}
    → multiply top & bottom with conjugate surds (C)
    {} = {(4-sqrt{5})(2sqrt{5}-3)}/{(2sqrt{5})^2 - (3)^2}
    {} = {8 sqrt{5} - 12 - 10 + 3 sqrt{5}}/{4(5)-9} = {11 sqrt{5} - 22}/{11} = sqrt{5} - 2

  2. Given that {2/sqrt{2}}({{3sqrt{54}}/4}-{15/sqrt{150}}-{14/sqrt{294}})=k sqrt{3}, find the value of k.

    Ans:

    → Rationalize all the sinful terms with surds in denominator!
    {2/sqrt{2}}*{sqrt{2}/sqrt{2}}({{3sqrt{54}}/4}-{15/sqrt{150}}*{sqrt{150}/sqrt{150}}-{14/sqrt{294}}*{sqrt{294}/sqrt{294}})
    {}=k sqrt{3}
    doubleright {{2 sqrt{2}}/2}({{3sqrt{54}}/4}-{{15 sqrt{150}}/150}-{{14 sqrt{294}}/294})=k sqrt{3}
    → Now simplify the surds using B(2) above and a tiny bit of trial and error
    doubleright sqrt{2}({{3sqrt{9*6}}/4}-{{15 sqrt{25*6}}/150}-{{14 sqrt{49*6}}/294})=k sqrt{3}
    doubleright sqrt{2}({{9sqrt{6}}/4}-{{sqrt{6}}/2}-{{sqrt{6}}/3})=k sqrt{3}
    doubleright sqrt{2}(17/12)sqrt{6}=k sqrt{3}
    → split √6 into √2√3 since we’re going to compare with the √3 term on RHS
    doubleright sqrt{2}(17/12)sqrt{2}sqrt{3}=k sqrt{3}
    doubleright (34/12)sqrt{3} = (17/6)sqrt{3} =k sqrt{3}
    doubleright k = 17/6 via comparing coefficient of √3

As always, get these rules drilled into your head! Spot the pointers and common mistakes in red! Understand the representative sample questions! Check out more surds in action!

Print this out if necessary and remember the above procedures by heart, for if you fail in rationalizing surds in your exams, you’ll need to write a 1000000-word essay to Sergeant Loi explaining the rationale for not punishing you!

*Cracks whip*

頑張って!!!

Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

  1. Leave A Comment!
Comments & Reactions

7 Comments

  1. krisandro's Avatar
    krisandro commented in tuition class


    2008
    May
    4
    Sun
    1:15am
     
    1

    Sergeant Loi is taking over the blog~ I prefer Miss Loi pls!

  2. |c3^sNoW's Avatar
    |c3^sNoW commented in tuition class


    2008
    May
    4
    Sun
    1:18am
  3. xinyun's Avatar
    xinyun commented in tuition class


    2008
    May
    4
    Sun
    10:43pm
     
    3

    hey miss loi, haha long time no visit the blog then got so much scary math stuff from your alter ego eh.
    oh btw i think i need saving!!!! for maths!!! : ( hahahaha

  4. ignorantsoup's Avatar
    ignorantsoup commented in tuition class


    2008
    May
    5
    Mon
    9:04pm
  5. Lim Ee Hai's Avatar
    Lim Ee Hai commented in tuition class


    2008
    May
    5
    Mon
    11:08pm
     
    5

    I rather solve 100 maths questions than to write 1 million word essay. I need to improve maths, not english yo...
    Ha ha!

  6. Sergeant Loi's Avatar
    Sergeant Loi commented in tuition class


    2008
    May
    6
    Tue
    11:37am
     
    6

    KRISANDRO!!! Sergeant Loi not appealing enough for you?! 1000000 word essay on her desk tomorrow morning!

    |c3^sNoW: Yup 1st victim claimed (see above)

    xinyun: You need saving again? Sure or not?! Thought all the stuff you've already learnt last year?

    P.S. For a moment, thought you're the famous xinyun on the internet.

    Soupie: You've just questioned the feminineness of Sergeant Loi's master! 1000000 essay on her desk tomorrow morning!

    Cher Lim: OK 1000000 maths questions for you! LOL

  7. |c3^sNoW's Avatar
    |c3^sNoW commented in tuition class


    2008
    May
    6
    Tue
    9:17pm
     
    7

    wahhh muz diam diam mai chut siah. until miss loi is back again. :p

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