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

Size: px

Start display at page:

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

Joseph Fields
5 years ago
Views:

1 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 simplify surds rtionlise surd denomintor simplify epressions using the lws of indices. Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

2 ALGEBRAIC OPERATIONS A. Reducing lgebric frctions to their simplest form Eercise. Simplify these frctions: () (b) 0 9 (c) p p (h) v w (i) rs rt (j) b c (k) d (l) m m (m) (n) 8z z (o) (p) d d (q) (r) v w (s) 8y (t) b (u) pq p (v) b b (w) yz z () ef 7ef (y) pq pq (z) 8y y () ( ) ( ) ( ) ( ) + + (b). Fctorise either the numertor or the denomintor, then simplify: () (b) (c) 8 9 (i) + y (j) pq + p p + (k) v + v v. Fctorise the numertor nd/or the denomintor, then simplify: + c + 9 () (b) (c) d + c + d (h) (l) g + g g + + y + y p+ q 7p+ 7q w w (h) y 8 8y (i) (j) y 9 y + (k) (l) w + 0 w 00 (m) + + (n) v v v (o) y + y y + y (p) Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

3 B. Multiplying, dividing, dding nd subtrcting lgebric frctions Eercise. Simplify these frctions by multiplying: () (b) (c) (h) (i) c b d (j) v (k) m m (l) b y w n n b (m) (n) (o) (p) n 7 (q) (r) d (s) c (t) c 9 n 9. Chnge these divisions to multiplictions nd simplify: () (b) 8 (c) d d m (h) m (i) (j) (k) (l) b b r r b b (m) (n) (o) (p) d d w w y d (q) b b (r) d d. Do the following dditions nd subtrctions: () + (b) + (c) Ð + Ð Ð + (h) Ð 7 (i) (j) (k) (l) 8 Ð c + d 0 (m) e Ð h (n) m Ð n (o) + k (p) u w 8 Ð (q) r + s (r) d (s) (t) Ð + y + u Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

4 . By finding common denomintor with letters, work out these dditions/subtrctions: () + (b) Ð (c) + Ð y b c d p q + Ð 7 + (h) Ð 8 v w g h k n y. Add or subtrct these frctions: () + + (b) (c) + + Ð Ð (h) Ð Ð Ð + + Ð Ð C. Chnging the subject of formul Eercise A This eercise hs mied selection of formule. Chnge the subject of ech formul to the letter shown in the brckets. ALL WORKING nd ALL STEPS SHOULD BE SHOWN.. + = c (). Ð = c (). + p = q (). Ð p = q (). / = (). / 7 = () 7. / y = () 8. / p = m () 9. / r = s () 0. = 0 (). = (). g = h (). n = t (). + = (). + = b (). + c = b () 7. + c = b () 8. p + q = r () 9. v Ð w = y () 0. D = S T (S). C = pd. = (). = y (). A = pr (r). T = D / S (S). A = y (y) 7. P = pr (p) 8. P = pr (r) 9. h Ð p = q (h) 0. h Ð p = q (p). h Ð p = q (h). h Ð p = q (p). b Ð c = () Eercise B. Chnge the subject of ech formul to h. () g = hf (b) e = g + h (c) k = h /f e = g Ð h. Chnge the subject of ech formul to r. () Q = r (b) N = pr (c) M = pr P = pr w Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

5 . Chnge the subject of ech formul to m. () A = klm (b) B = Km (c) C = pmr D = pm. Chnge the subject to. () p = q + (b) r = s Ð (c) r = s Ð r = 7 Ð m = ( + ) m = / ( Ð ) n = / ( + ) (h) p = / ( + q). Chnge the subject of the formul to the letter in brckets. () P /Q = R (P) (b) t = /s (s) (c) M = P /Q (Q) v d e = f K mn = T (n) R w = (w) z = 7 (s) (h) + b = c () 9s. Hrder emples. Chnge the subject of the formul to the letter in brckets. () A + d = V /T (T) (b) p + q = r () (c) = b + c () m = r Ð s (s) = v Ð w (w) p = Ör Ð (r) s v + w D. Simplifying surds Eercise. Epress ech of the following in its simplest form: () Ö8 (b) Ö (c) Ö7 Ö0 Ö0 Ö8 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 subtrct the following: () Ö + Ö (b) Ö Ð Ö (c) 8Ö0 + Ö0 9Ö0 Ð 9Ö0 Ö Ð Ö Ö + Ö Ð Ö Ö7 Ð 8Ö7 + Ö7 (h) 0Ö + 0Ö. Simplify: () Ö8 Ð Ö (b) Ö8 Ð Ö (c) Ö + Ö Ö8 + Ö Ö + Ö0 Ö Ð Ö8 Ö0 + Ö8 (h) Ö7 Ð Ö Eercise. Simplify: () Ö Ö (b) Ö Ö (c) Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö (h) Ö Ö (i) Ö Öc (j) Ö Öy (k) Ö Ö8 (l) Ö Ö (m) Ö Ö (n) Ö0 Ö0 (o) Ö Ö (p) Ö Ö contd. Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils 7

6 . Multiply out the brckets: () Ö( + Ö) (b) Ö(Ö + ) (c) Ö(Ö + ) Ö7(+ Ö7) Ö( Ð Ö) Ö( Ð Ö) (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 = + Ö nd b = Ð Ö, simplify: () + b (b) b (c) + b. If r = Ö + Ö nd s = Ö Ð Ö, simplify: () r Ð s (b) rs (c) r Ð s. A rectngle hs sides of length (Ö + ) cm nd (Ö Ð ) cm. Clculte: () its re (b) the length of digonl Ö + Ö Ð E. Rtionlising surd denomintor Eercise. Rtionlise the denomintors in the following nd simplify where possible: () (b) (c) (i) (m) 7 0 (j) (n) 0 0 (k) (o) 0 (h) (l) (p) 8. Epress ech of the following in its simplest form with rtionl denomintor: () (b) (c) 0. Simplify the following by rtionlising the denomintor: b () (b) 7 (c) (h) 9 Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils 8

7 F. Simplifying epressions using the lws of indices Eercise 7 Rule m n = m+n. Use Rule to write down the simplest form of the products in the following: () (b) (c) b b c c (h) d d (i) v v 8 (j) 0 (k) w w (l) z z (m) f f 7 (n) g 0 g 0 (o) k k (p) m 00 m Rule m n = mðn. Use Rule to write down the simplest form of the quotients in the following: () (b) (c) b b c 9 c (h) d d (i) v v (j) 0 (k) w w (l) z z (m) f 7 f (n) g 0 g 0 (o) k k (p) m 00 m (q) (r) m (s) (t) r 9 m r 8 Rule ( m ) n = mn ( ) (8 ) u 7 u. Simplify these. For emple, ( ) =. () ( ) (b) (y ) (c) (z ) (g ) 8 ( ) 7 (b ) (c ) (h) (d ) 7. Epress the following without brckets, writing nswers in inde form: () (b) (7 ) (c) ( ) ( 7 ) 7 ( ) (9 ) (h) ( ). Note (b) m = m b m Use this result to simplify: () (b) (b) (cd) (c) ( y) 0 (pq). Use the rules lerned so fr to simplify the following: () y y (b) t t (c) v 7 v (h) 8 (i) (j) ( + ) (k) ( Ð ) (l) (uv) 7 (m) (y) (n) (mn ) 8 (o) (p) u Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils 9

8 7. Find m. For emple: if m = 8, m = since = 8. () m = (b) m = (c) m = m = Eercise 8 Rule 0 =. Write down the vlues of: () 0 (b) 0 (c) 0 () 0 Rule Ðm = / m. Write the following with positive indices. For emple: Ð = /. () Ð (b) Ð7 (c) Ð b Ð9 Ð y Ð y Ð (h) (i) t (j) c (k) y (l). Epress these in form without indices: () 0 (b) 7 Ð (c) Ð 8 Ð. Simplify the following. For emple: Ð = Ð+ =. 7 æ ö è ø () Ð (b) b b Ð (c) c Ð c Ð d Ð d e e Ð g Ð g w Ð w Ð (h) ( Ð ) (i) (y ) Ð (j) (z Ð ) Ð (k) (klm) 0.. Epress with positive indices: () Ð (b) Ð Ð (c) 7 Ð Ð ( ) Ð (9 0 ) Ð8 (w Ð ) Ð (h) (i) (j) (k) h Ð (l) 9s Ð (m) k (n) m. Multiply out the brckets: () ( + Ð ) (b) ( Ð Ð ) (c) Ð ( + ) Ð ( + ) ( Ð Ð7 ) Ð ( Ð ) Ð ( Ð ) (h) ( Ð Ð ) 7. () Find the vlue of:, 0, Ð,, Ð. (b) Write in the form Ð p : /, /, /. Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils 0

9 Eercise 9 Rule m/n = n Ö m. Write these in root form. For emple: = () (b) m (c) r w n r. Write these in inde form. For emple: Ö = / () (b) b (c) z w. Evlute the following. For emple: 8 = 8 = () = () 9 (b) (c) (h) (i) 9 (j) (m) 8 (n) 8. Simplify: () (o) æ ö è ø (p) æ ö è 8ø (k) 7 (l) ( ) (b) m 9 ( ) (c) c ( ) ( n ) u ( ) n æ g ö ç è ø æ ö çb è ø æ (h) z ö ç è ø (i) æ ç è ö æ (j) ç ø è ö (k) + ø ( ) (l) + + ( ) Eercise 0 Miscellneous emples. Simplify: () e (b) b b e æ ö ç z è ø (c) c c æ çw è ö ø d d (h) æ w ç è ö ø 0. Simplify: () / Ð/ (b) b / b / (c) c / c / d / d Ð/ e / e / 8v / v Ð/ z Ð/ z Ð/ Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

10 . Multiply out the brckets: () / ( / Ð Ð / ) (b) / ( / + / ) (c) Ð/ ( / + Ð/ ). Evlute the following for = nd y = 7 : () / (b) / (c) y / 8y Ð/ Ð/ y /. Simplify the following, epressing your nswers with positive indices: () p p / (b) p p Ð/ (c) (p / ) Ð/ (p Ð/ ) p / p Ð/ 7p / 7p Ð/. Simplify: () (b) (c) 7. Multiply out the brckets: () ( Ð )( Ð Ð ) (b) ( Ð + )( Ð Ð ) (c) ( / + )( Ð/ Ð ) Eercise Epress the following with in the numertor in inde form: For emple : = = = m m ( ). ( ) Ð Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils Ð/ /

11 Checkup for lgebric opertions Ð/ / Ð/. Fctorise the numertor nd/or denomintor if possible, then simplify: () 7v 9 (b) b v (c) b Ð 8 + y Ð 8 v + v + + 9y Ð v Ð v Ð vw v. Simplify: () b (b) (c) z z b b b c (h) c u v Ð + Ð 8 k m. Chnge the subject of ech formul to. () + = (b) Ð = w (c) + m = p / z = p /w N = p T = + w = / ( + y) (h) M = 8. Epress ech of these in its simplest form: () Ö (b) Ö000 (c) Ö Ö Ð Ö0 9 Ö0 Ð Ö8 Ö7 + Ö0 (h) Ö8 Ð Ö. If = + Ö nd y = Ð Ö, simplify: () + y (b) y (c) + y. Rtionlise the denomintors in the following nd simplify where possible: () /Ö (b) 8 / Ö (c) /Ö Ö/ Ö 7. Write in their simplest form: () 8 (b) 8 (c) ( ) ( b) (p ) 0 (h) ( Ð ) 8. Write with positive indices: () Ð (b) b Ð (c) (y Ð ) Ð 9. Write these in root form: 0. Write these in inde form: () b / (b) c Ð/ () Ö (b) / Ö. Evlute: () / (b) Ð/ (c) ( 8 ) / (y 7/ ) Ð Ð/ ( / Ð Ð/ ) s / s Ð/ Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

12 ANSWERS Algebric opertions Eercise. () / (b) / (c) / / / / / (h) v /w (i) s /t (j) b /c (k) /d (l) /m (m) / (n) 8 /z (o) (p) /d (q) / (r) v / w (s) y (t) / b (u) q / (v) /b (w) y () / 7 (y) / q (z) /y () /( + ) (b) /( Ð ). () + (b) + (c) Ð Ð + (h) (i) Ð y (j) q + Ð + Ð (k) v + (l) Ð. () (b) (c) / g / / 7 (h) /8 (i) + (j) y Ð (k) + (l) Ð v + y + /(w Ð 0) (m) + (n) v + (o) y + (p) Ð + Eercise. () / (b) / 9 (c) / 7 / (h) / (i) c /bd (j) v /yw (k) m (l) (m) (n) (o) (p) (q) n 0 7 /d (r) (s) (t). () / (b) / (c) / / / / (h) (i) (j) b / (k) r / (l) (m) d (n) / w (o) / (p) d / y (q) (r) /. () 7 / (b) / (c) / 0 8 / / / 7 / 0 (h) / 0 (i) / 8 (j) 9 / 0 (k) + (l) c + d (m) e Ð h (n) m Ð n (o) + k 0 8 (p) u Ð w (q) 8r + s (r) Ð d (s) + 9y (t) + 8u () y + (b) b Ð (c) d + c q Ð p y b cd pq w + v vw Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils 9

13 h Ð g gh 7n + k kn (h) y Ð 8 y. () + (b) (c) + 7 Ð (h) Eercise A. = c Ð. = c +. = q Ð p. = q + p. =. = 7 7. = y 8. = mp 9. = rs 0. =. = /. = h /g. = t /n. =. = b Ð. = b Ð c 7. = b Ð c 8. = r Ð p q 9. = y + v w 0. S = D /T. d = C /p. = or Ð. = 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. = b Ð c Eercise B. () h = g /f (b) h = e Ð g (c) h = kf h = g Ð e. () r = ÖQ (b) r = Ö( N /p) (c) r = Ö( M /p) r = Ö( P /pw). () M = A /kl (b) m = B /K (c) m = c /pr m = /pd. () = p Ð q (b) = s Ð r (c) = (s Ð r) / = (r + ) /7 = (m Ð ) / = m + = n Ð (h) = p Ð q. () P = QR (b) s = /t (c) Q = Ö( P /M) w = v z f = e /d n = K /mt s = Ö( 7 /9r) (h) = Ö(c Ð b ). () T = V / (+d) (b) = r /(p + q) (c) = c /( Ð b) = r /(m + ) w = (v Ð v) / ( + ) r = (p + ) Eercise. () Ö (b) Ö (c) Ö Ö Ö Ö7 Ö (h) Ö (i) 0Ö (j) Ö (k) Ö (l) Ö (m) 0Ö (n) 7Ö (o) Ö (p) Ö (q) 0Ö (r) Ö0 Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils 0

14 . () 8Ö (b) Ö (c) Ö0 0 ÐÖ ÐÖ 0 (h) 0Ö + 0Ö!!. () Ö (b) Ö (c) 0Ö Ö Ö Ö7 8Ö (h) Ö Eercise. () (b) (c) Ö Ö (h) Ö (i) Ö(c) (j) Ö(y) (k) (l) 8 (m) Ö (n) 0Ö (o) (p) Ö. () Ö + (b) Ö + Ö (c) + Ö Ö7 + 7 Ö Ð Ö Ð 8 + Ö (h) ÐÖ (i) (j) (k) + Ö (l) 8 Ð Ö. () (b) Ð (c). () Ö (b) 0 (c) Ö. () cm (b) Ö cm Eercise. () Ö / (b) Ö / (c) Ö / Ö / Ö7 /7 Ö Ö / (h) Ö / (i) 0Ö (j) Ö (k) Ö (l) Ö /0 (m) Ö / (n) Ö /0 (o) Ö /0 (p) Ö. () Ö / (b) Ö0 / (c) Ö0 / Ö / Ö /. () Ö + (b) (Ö7 + ) / (c) ( Ð Ö) / Ö Ð + Ö Ö + Ö Ö Ð Ö (h) Ö + Ö Eercise 7. () 7 (b) 8 (c) b 8 c 8 (h) d 0 (i) v (j) (k) w (l) z (m) f (n) g 0 (o) k (p) m 0. () (b) (c) b c (h) (i) v (j) 8 (k) w (l) z (m) f (n) (o) k 0 (p) m 99 (q) (r) m (s) (t) r. () (b) y (c) z 0 g b c 0 (h) d. () 0 (b) 7 (c) (h). () b (b) c d (c) 0 y 0 p q. () y (b) t (c) v 7 (h) (i) (j) + (k) Ð (l) u 7 v 7 (m) y (n) m 8 n (o) (p) u 7. () m = (b) m = (c) m = m = Ö(b) b Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

15 Eercise 8. () (b) (c). () / (b) / 7 (c) / /b 9 / /y /y (h) (i) t (j) c (k) / y (l) / 7. () (b) / 7 (c) / 7 / 8 9 /. () (b) b (c) c Ð e 9 g Ð8 w (h) Ð (i) y Ð (j) z (k). () / (b) / (c) / 8 / w (h) (i) (j) 0 (k) /h (l) 9 /s (m) /k (n) /m. () 7 + (b) Ð (c) + Ð + Ð Ð Ð Ð Ð Ð (h) Ð 7. () 7,, / 9,, / 7 (b) Ð, Ð, Ð Eercise 9. () Ö (b) Öm (c) Ör Öw Ön /( Ör ). () / (b) b / (c) z / w / Ð/ u /. () (b) 8 (c) 0 7 (h) 8 (i) (j) / (k) / (l) / (m) 7 (n) /7 (o) (p) /. () (b) m (c) c n Ð n Ð/ g b Ð (h) z (i) / (j) (k) (l) Eercise 0. () (b) b (c) d e / z w Ð (h). () (b) b (c) c d e v. () Ð (b) + (c) / + Ð/. () (b) (c) 8 / 9 /. () p / (b) p / (c) /p /p 9 p / 9 /p. () (b) (c) Ð9/ 7. () Ð Ð / (b) / Ð (c) Ð / + / / Ð Eercise. Ð. Ð. 7 Ð. Ð. / Ð. / Ð 7. / Ð 8. Ð/ 9. / Ð/ 0. /. Ð. Ö Ð / Ö Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

16 Checkup for lgebric opertions. () / (b) /b (c) /( Ð ) v Ð w + y /( + 8) (v + ) + y /(v Ð ). () b / (b) (c) z c /8 (u Ð 8v) /0 (m + k) /km (h) ( + ) /0. () = (b) = w + (c) = (p Ð m) / = pz /w = Ö( N /p) = (T Ð ) / = w Ð y (h) = Ö( /8M). () Ö (b) 0Ö0 (c) Ö Ö Ö Ö / (h) Ö. () (b) Ð0 (c) 8. () Ö / (b) Ö (c) Ö Ö / 7. () (b) (c) m b (h) Ð 8. () / (b) /b (c) y ( )/ 9. () Öb (b) /Öc 0. () / (b) Ð/. () (b) / (c) /y 7 / Ð / s / Qudrtic functions Eercise. () y = (b) y = (c) y = / y = Ð y =Ð / y = Ð. () y = + (b) y = + (c) y = Ð y = Ð + y = Ð + y = Ð Ð. () y = ( Ð ) + (b) y = ( Ð ) + (c) y = ( Ð ) y = ( + ) + y = ( + ) y = ( + ) Ð y = ( Ð ) + (h) y = ( Ð ) Ð (i) y = ( + ) Ð Eercise. () (,) (b) = (c) y = ( Ð ) +. () (i) (,) (ii) = (iii) y = ( Ð ) + (b) (i) (,Ð) (ii) = (iii) y = ( Ð ) Ð (c) (i) (Ð,) (ii) = Ð (iii) y = ( + ) + (i) (Ð,) (ii) = Ð (iii) y = ( + ) + (i) (,Ð) (ii) = (iii) y = ( Ð ) Ð (i) (Ð,Ð) (ii) = Ð (iii) y = ( + ) Ð. () (, ); = (b) (,7); = (c) (8,); = 8 (Ð,); = Ð (,Ð); = (Ð,Ð7); = Ð (,0); = (h) (Ð,0); = Ð (i) (0,); = 0 Mthemtics Support Mterils: Mthemtics (Int ) Ð Student Mterils

ALGEBRAIC OPERATIONS. A. Reducing algebraic fractions to their simplest form

$ALGEBRAIC OPERATIONS. A. Reducing algebraic fractions to their simplest form$ ALGEBRAIC OPERATIONS A. Reducing lgebric frctions to their simplest form Students should be shown the method for Ôcncelling downõ vulgr frction, then frctions with letters cn be introduced. * Before this