It should be emphasised that ALL WORKING and ALL STEPS should be shown.

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1 C. Changing the subject of a formula It should be emphasised that ALL WORKING and ALL STEPS should be shown. The following examples could be used to introduce changing the subject of a formula. The examples are a mixture of Ôchange side, change signõ, cross multiplication, dividing and finding the square root. The examples are a mixture of Ôchange side, change signõ, cross multiplication, dividing and finding the square root. Example Change x + = m to x. Example Change a /b = p to b Ans. x + = m Ans. a / b = p x = m Ð bp = a (cross x) b = a /p Example Change ax Ð c = d to x. Example Change A = pd to d Ans. ax Ð c = d Ans. A = pd ax = d + c pd = A x = d = d + c A /p a d = A / p Question of Exercise provides harder examples, this example could be used as extension: Example: Change y = v Ð z z to z. v Ð z Ans. y = z zy = v Ð z zy + z = v z(y + ) = v z = v y + Exercise may now be attempted. Ö D. Simplifying surds Irrational numbers could be introduced by first revising sets of numbers. E.g. Real nos. - all the numbers which can be represented on a number line. Whole nos. 0,,,,,,,... Integers...-, -, -, 0,,,,,... Rational nos. 8, -, /, -/ etc. numbers which can be expressed as a fraction. Then explaining that numbers like Ö, Ö, p... cannot be expressed as a fraction, therefore these are irrational. contd. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

2 A SURD is a special kind of irrational number. It is a square root, a cube root, etc. which cannot be expressed as a rational number. Ö, Ö, Ö0 are all surds, whereas Ö and Ö8 are not surds as Ö = and Ö8 =. The following examples could be used to show students how to simplify surds: Example Express Ö8 in its simplest form. Ans. Ö8 = Ö(9 x ) = Ö Explain why Ö8 = Ö( x ) is not used. Largest square number which divides into 8 Example Simplify Ö8 + Ö Ans. Ö8 + Ö = Ö( x ) + Ö = Ö + Ö = 7Ö Exercise may now be attempted. Example Simplify Ö x Ö Example Simplify Ö x Ö Ans. Ö x Ö = Ö = Ans. Ö x Ö = 0Ö8 Example Simplify Ö( Ð Ö) = 0Ö(9 x ) Ans. Ö( Ð Ö) = 0 x Ö = Ö Ð ÖÖ = 0Ö = Ö Ð (x) = Ö Ð Exercise may now be attempted. E. Rationalising a surd denominator Students should be reminded of the difference between a numerator and a denominator and an explanation given of what rationalising a denominator means. Example Express with a rational denominator. Ö Ans. can be multiplied by, without changing its value. Ö The number ÔÕ can be written here as Ö. Ö So x Ö will have the same value as but will be written differently. Ö Ö Ö x Ö = Ö = Ö Ö Ö Example Express 9 with a rational denominator. Ö 0 Ans. 90 = = x Ö0 = Ö0 Ö Ö0 Ö0 Ö0 0 Exercise Q and Q may now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

3 Exercise Q contains eight examples appropriate to grades A/B. Example Express + Ö with a rational denominator. Ans. Here, multiply by Ð Ö to rationalise the denominator. Ð Ö So, Ð Ö = Ð Ö x = Ð Ö + Ö Ð Ö Ð Rule = a mn Notice - no Ö term in denominator Exercise Q may be attempted now (for extension to grades A/B). F. Simplify expressions using the laws of indices Basically, there are rules for the students to learn. They should be lead through them, doing examples of each type, before attempting Exercise 0, containing miscellaneous examples. The rules are: Rule a m x a n = a m+n Rule a m a n = a mðn Rule (a m ) n = a mn Rule a 0 = Rule a Ðm = /a m Rule a m/n = n Öa m Examples Rule 7 x = 7 + = x x x = 0x + = 0x 7 Rule Rule 7 = 7 Ð = ( ) = x = a 8 a = a 8 Ð = a (x y ) = x x y x = x 8 y Exercise 7 may now be attempted. Rule Rule Ðve power = /+ve power (Any number) 0 = Ð = / = /9 (0) 0 = / xð = //x = x Exercise 8 may now be attempted. a (a Ð a Ð ) = a + Ð a Ð = a Ð a Ð = a Ð /a Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 7

4 . Change the subject of each formula to m. (a) A = klm (b) B = Km (c) C = pmr (d) D = pm. Change the subject to x. (a) p = q + x (b) r = s Ð x (c) r = s Ð x (d) r = 7x Ð (e) m = (x + ) (f) m = / (x Ð ) (g) n = / (x + ) (h) p = / (x + q). Change the subject of the formula to the letter in brackets. (a) P /Q = R (P) (b) t = /s (s) (c) M = P /Q (Q) (d) v (e) d e = (f) (f) f K mn = T (n) (g) R w = (w) z = 7 (s) (h) a + b = c (a) 9s. Harder examples. Change the subject of the formula to the letter in brackets. (a) A + d = V /T (T) (b) px + qx = r (x) (c) ax = bx + c (x) (d) m = r Ð s (s) (e) x = v Ð w (w) (f) p = Ör Ð (r) s v + w D. Simplifying surds Exercise. Express each of the following in its simplest form: (a) Ö8 (b) Ö (c) Ö7 (d) Ö0 (e) Ö0 (f) Ö8 (g) Ö8 (h) Ö (i) Ö00 (j) Ö7 (k) Ö (l) Ö7 (m) Ö00 (n) Ö7 (o) Ö (p) 7Ö8 (q) Ö (r) Ö0. Add or subtract the following: (a) Ö + Ö (b) Ö Ð Ö (c) 8Ö0 + Ö0 (d) 9Ö0 Ð 9Ö0 (e) Ö Ð Ö (f) Ö + Ö Ð Ö (g) Ö7 Ð 8Ö7 + Ö7 (h) 0Ö + 0Ö. Simplify: (a) Ö8 Ð Ö (b) Ö8 Ð Ö (c) Ö + Ö (d) Ö8 + Ö (e) Ö + Ö0 (f) Ö Ð Ö8 (g) Ö0 + Ö8 (h) Ö7 Ð Ö Exercise. Simplify: (a) Ö x Ö (b) Ö x Ö (c) Ö x Ö (d) Ö x Ö (e) Öx x Öx (f) Ö x Ö (g) Ö x Ö (h) Ö x Öa (i) Ö x Öc (j) Öx x Öy (k) Ö x Ö8 (l) Ö x Ö (m) Ö x Ö (n) Ö0 x Ö0 (o) Ö x Ö (p) Ö x Ö contd. Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 7

5 . Multiply out the brackets: (a) Ö( + Ö) (b) Ö(Ö + ) (c) Ö(Ö + ) (d) Ö7(+ Ö7) (e) Ö( Ð Ö) (f) Ö( Ð Ö) (g) (Ö + )(Ö Ð ) (h) (Ö Ð )(Ö + ) (i) (Ö + Ö)(Ö Ð Ö) (j) (Ö Ð Ö)(Ö + Ö) (k) (Ö + Ö) (l) (Ö Ð Ö). If a = + Ö and b = Ð Ö, simplify: (a) a + b (b) ab (c) a + b. If r = Ö + Ö and s = Ö Ð Ö, simplify: (a) r Ð s (b) rs (c) r Ð s. A rectangle has sides of length (Ö + ) cm and (Ö Ð ) cm. Calculate: (a) its area (b) the length of a diagonal Ö + Ö Ð E. Rationalising a surd denominator Exercise. Rationalise the denominators in the following and simplify where possible: (a) (b) (c) (d) (e) (i) (m) 7 0 (f) (j) (n) 0 0 (g) (k) (o) 0 (h) (l) (p) 8. Express each of the following in its simplest form with a rational denominator: (a) (b) (c) 0. Simplify the following by rationalising the denominator: (d) (e) (f) a b (a) - (b) 7 - (c) + (d) + (e) - (f) - (g) + (h) 9 - Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 8

6 (f) h Ð g gh (g) 7n + k kn (h) y Ð 8x xy. (a) x + (b) 9x + 0 (c) x + 7 (d) x Ð (e) (f) x (g) x + (h) x x + Exercise A. x = c Ð. x = c +. x = q Ð p. x = q + p. x = a. x = 7a 7. x = ya 8. x = mp 9. x = rs 0. x =. x = a /. x = h /g. x = t /n. x =. x = b Ð. x = b Ð c 7. x = b a Ð c 8. x = r Ð p q 9. x = y + v w 0. S = D /T. d = C /p. x = or Ð. x = Öy or ÐÖy. r = Ö( A /p). S = D /T. y = ÖA (or ÐÖA) 7. p = P /r 8. r = Ö( P /p) 9. h = q + p 0. p = h Ð q. h = q + p. p = h Ð q. x = b a Ð c Exercise B. (a) h = g /f (b) h = e Ð g (c) h = kf (d) h = g Ð e. (a) r = ÖQ (b) r = Ö( N /p) (c) r = Ö( M /p) (d) r = Ö( P /pw). (a) M = A /kl (b) m = B /K (c) m = c /pr (d) m = /pd. (a) x = p Ð q (b) x = s Ð r (c) x = (s Ð r) / (d) x = (r + ) /7 (e) x = (m Ð ) / (f) x = m + (g) x = n Ð (h) x = p Ð q. (a) P = QR (b) s = /t (c) Q = Ö( P /M) (d) w = v z (e) f = e /d (f) n = K /mt (g) s = Ö( 7 /9r) (h) a = Ö(c Ð b ). (a) T = V / (a+d) (b) x = r /(p + q) (c) x = c /(a Ð b) (d) x = r /(m + ) (e) w = (v Ð vx) / (x + ) (f) r = (p + ) Exercise. (a) Ö (b) Ö (c) Ö (d) Ö (e) Ö (f) Ö7 (g) Ö (h) Ö (i) 0Ö (j) Ö (k) Ö (l) Ö (m) 0Ö (n) 7Ö (o) Ö (p) Ö (q) 0Ö (r) Ö0 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 0

By the end of this set of exercises, you should be able to. reduce an algebraic fraction to its simplest form

$By the end of this set of exercises, you should be able to. reduce an algebraic fraction to its simplest form$ ALGEBRAIC OPERATIONS By the end of this set of eercises, you should be ble to () (b) (c) reduce n lgebric frction to its simplest form pply the four rules to lgebric frctions chnge the subject of formul