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Contents Questions Set Number Basic Indices Review of Index Laws Negative Indices Scientific Notation Answers Set Number Basic Indices Review of Index Laws Negative Indices Scientific Notation
f fe D, 0 m--, (as :- li,,_,. 5 e n z..," u - k_,, e Do marked questions (as per teacher instruction, plus any others by teacher request) E These are extension questions Set Basic Indices (key integer bases) E,Le re-5s n C i k,c_ fv) a (cx) Dc. X. Cb) - e ( C) 'S - a (el) (7'0 (!..) ( ek)t pk)._..._.. tka. tett-c.. (uotili o o, coszaiaor f)ost6-0,..) - S d r 5 7 (a) ' 7-i... 6) (i0),- ) ; - ) - ))) (c ) 5 I)-sq", 5 0 Ca) e 5 7 5-/55 ( ) Ca di () ;-/- (k) a'.) 0 ) 0'), Paq-, lo s ) IdG fp co I" p /, 9 6 (. 6. U rtk_e_ Jt st;yt, ( 0,) (a-) g - khave...x. ( 6 - ) - i p o i orzso * (s) 0 t>oo/octia O. to 'MAL. -et& u;c -( L: ) tiail `AS& " COCCADO:tOr fc)ssamd- _ 7_ ( a) (-- ) (-- -)5 (- ) (a.) (..Y * L b / Cs-) (-5) * - (eo (70 0 (C.) (77 ) - EB) 0-
Set Review of Index Laws To simplify expressions with constants and/or pronumerals in index form, the following index laws are used.. a n, x a n = n. a"' a = a' n n. a =. (when a 0). (a)" amn 5. (abr = m 6. ] Review' 7 index laws 6 77 Wir' n i 777.7tWaMWATAWORAMTZWAWRZTIRMIalTgligNATARKW,AgNMN 0 Simplify each of the following. a a x a 0 b a x a x a rj c bxb5x.7 d ab x a b 5 e / /l 6 X / n 7 f a b 5 c x abc o g mnp x m 5 n p h a x ab I a b x 5a b x j m x rnn x 6m n 5 0 k x x x 6x y 0 I x y x x x xy OriFti) Simplify each of the following. a a a b a 7 a d a 7 g 0 e b6 a 7b m 7 n x y m ej 7 ab5c h.,cy nm5 n p ab c k - 6/np c b6 b 8m8 f m 6x 7 y 8x xyz 8x y z
C.) ei7:) Simplify each of the followina 0 a a 9 b (M) o d x 0.o e b 0 g a h 5y CeLa Simplify each of the following. a (ay d (- g (/nn5) j c 5J m (--)5 b (a5) c h ( mn k 7x ) y5 n (-7) 5 a? a in 0 n 5 is the simplified form of: A m 5 n x mn D 0. 5 ) x b The value of A D (5a) 0 is: o c (m)0 f x (n) et i 5x 0 - (5xy)0 in` e (ab) 9 f (ab) 6m 0 n n 5 \ (rn n B 9 E 5 6 CB a a b x b x 5a b simplifies to: A 9a 5 b 8 B 0a5b7 D 9a 5 b 7 E a5b8 i lanz b ) a ) 5b o (-)5 C C (/5n) C 0a5b8 5x 9 X x6 simplifies to: 9x 0 x x A 5x 9 a 9x 9x 9 E 5x p 7 x 8q9 simplifies to: p x q A q B p q D P q q 7 a 5 b 7, a 5a6 b 5b5a simplifies to: 9a b 5a b A 5 9 F 5ab D ab - 9 C 5X9 - C ab
7 Evaluate each of the following, 0 a x x 0 c d 5 x 6 6 e ( x 5) x f g x 56 x 55 0 h ( x ) 8 Simplify each of the following. e a (X Y ) b a h x (pq)0 n 0 d a e n y mq (5) v5 i (5 x 5)0 C m a X x (inn) f (a m + E 9 Find algebraically the exact value of x if x+ I = x Justify your answer. 0 Binary numbers (base numbers) are used in computer operations. As the name implies, binary uses only two types of numbers, 0 and, to express all numbers. A binary number such as 0 (read one, zero, one) means ( x ) + (0 x ) + ( x ) = + 0 + 5 (in base 0, the base we are most familiar with). The number 00 (read one, zero, one, zero) means ( x ) + (0 x ) + ( x ) + (0 x ) = 8 + 0 + + 0 = 0. If reading the binary number from right to left, the index of increases by one each time, beginning with a power of zero. Using this information, write out the numbers to 0 in binary (base ) form. AT.- 5 5.5\ Why are these laws called index laws?
Set Negative Indices. A term with a negative index can be expressed with a positive index using the Seventh Index Law. (a) an (b) a-n = an. All index laws apply to terms with negative indices.. Always express answers with positive indices unless otherwise instructed.. Numbers and pmnumerals without an index are understood to have an index of. Negative indices <Via A it :;:k 'Ar, - ilkeuo 7.7:7,70WitMA MIV ig zte g MeAt' :. '' : 5:Wii'a=natIgk',I, ::; l I OM Express each of the following with positive indices. O a X -5 0 b y - 0 c a-9 O d la - 0 e.c y,0 f -m-/- 0 g 6a b - c -5 0 h a i a 6a, 7a - m -5 0 j 0 K --- 0 I F b a-b 0 Ilift5' Simplify each of the following, expressing the answers with positive indices. c a a b - X a -5-0 b x - y x y 0 c m n -5 x m-n- O d a b a 5 b 7.0 e xy 6 x y 5 0 f 5x - y 6xy 6m n x 9 h v M/- 0. g 0 0 i n M 6 X 7 y - 6m5/7 O Oj (a m ) -5 0 m ( - - q j I k (p 7 q - ) - 0 (a--) e n ( a ) F AligET - Evaluate each of the following without using a calculator. a a - b 6 - ia c - O d - x p e - X d f 5 x 6- - 0 0 a 6 (- b-j \- 6 x - I - e g --, 0 h 0 - x 5 x 9--' - 6 x 0 k 5 x5 x 8 X - 5 x 5 - x I 5 O j 0 O Evaluate each of the following, using a calculator. Oa -6 lb - -7 d (-) -a 0 e S ta x-- 5 is the same as: A -x 5 I -5x C 5x c 7-5 e f (0.05)-5 D -5 x E -5 X
I b is the same as: a - A a a. c - is the same as: 8 A -a E -a B - C a r- D - in 6 D a Which of the following, when simplified, gives 7? n- - /7/ n - D -l in- A a 6 b a b - B x - x rn X - E m x L n 9b a6 a E b. ". - b When simplified, a-/- a - b 6 is equal to: c When (x6y-`')- is simplified, it is equal to: 9a C b x 8 x 8 Y A B C Y 8y - 8x 8 D 8y x I 8 E x 8 6y x \ a x 8b9 d If -- is equal to 6. -, then x and y (in that order) are: ` b y } a n - - and -6 B -6 and - C - and D - and - E - and - - r w%' -A-,-, m - ' -tia ' - v,,, M=arigr- r5ommtit,w*0 M.0=ivv,.. 0A-r-M--- l i ff7.,,.,.%' j '. `.' ' '' -- -. ' -', -- e..'. -,, 77,,f:, n,..,..,. F,W,,,, exif,, - vg: Rst,A /, --k- -&-N-,--,.. E 7 Simplify, expressing your answer with positive indices. - - -.7 in n ((inn ) - E a b m 5 //6 (tn -5 n ) 5(a b - ) (5a-b)-I E c : (C) - T (a- b) 8 Simplify, expanding any expressions in brackets, a (r s ) (r S ) b (m 5 + n5) (x a + ) xx a+b I x+f - 8(x +) p P P a(b ) b x- x x x X X \. P ( P ) x 0 ( P )
Set Scientific Notation (Standard Forml Express each of the following as a basic numeral. a 9.8 x ICA 0 b 6.5 x.0 O c.5xw o d 7. x LT* O e x t i g's 6 f x t0-' a g.07 x G 0h 5.0 x 0-8.6 x 0' 0 j.8 x I 0-* k 7 x 0 g. *. x.06 o. Express each of the following in standard form. 0 a 6 000 b 560 o c 90 00 a d 8 050 000 e 0.0 o 0.000 O g 56 000 000 o h 90 000 6 0.000 07 66 j 0.000 000 5 8 k 0.000 0. 0.05 Express each quantity described below in standard form_ 0 a The diameter of a nucleus of an atom is about 0.000 000 000 000 0 in. ton Aron nucleus electron 0 b The highest Sherpa village on the slopes of Mount Everest is 000 m above sea-level.
c This micrograph shows a blood capillary in skeletal muscle. These fine capillaries which carry blood to the body's tissues are about 0.000 0 m in diameter. d The filament of a light bulb reaches a temperature of 70 C. e e The largest recorded Australian sporting crowd of nearly 700 attended the 970 Victorian Football League Grand Final. The Sun is approximately 50 000 000 km from Earth. The temperature at the surface of the Sun is about 6000 C.
B Ick _tcs). 5 ta.) )0 C6.) ( C - ) (:0 ( 6 - ) ).` ' ) = ( gv O cl- cjr- Z s,. 6, ' = t. (c) S i 7-9 ) /7 ( --. 6 ( ) cr I 0 Lk-' _5( C- ) 5 s (-r) = G fc;l- () =, S./ =_.5 S- 6S -' _6 0,- 5 z_ (-0 9- '8, =7-9 ((I) t ol = to tiz)-- = 00 (S (000 5,.---.. to op:do k = I Do, C, t0 cso 0 (0 0 -) t Oc.) _5 (k) = C)- (a) t. (-Z.) to 0 coo (0 5 kf) top oo coo to7.. It")- C o, ) ( -D (c_) )5 C-5) (L c) C-) ) (-5) LS 0,.,) (-6) / (
Exercise IA Review of index laws a a 7 b a 6 C ba m 5 n I f a5b7c g m6 p5 i 0a b j 9 6m 8 n 7 k X5y6 a a b a 5 C b e b f m 5 g mit x y 76 k.,0 a b c d e f - g h -7 i a a 6 b 6a c - I nas st d n 8 e a 6 b f 9a6b 9 h. 7 /7 6 n a 6 65m x8a k I 5 8 8y 65bI g 6m / i m - n 9 ci - 5 a D b D 6 a C b E c B d D 7 a 6 b 7 c 65 d8 e600 f 7 d ab7 h 6ab x8y6 d ±a h y Xy g0 hi i 8 a xvz b a b c manb x d a e -Prn-q f CP ' " ba- a! b c! d S 6 8 9 e [ f g 8 h 6 6 7 7 = k5 I 5 5 a 0.00 7 7 b 0.000 08 5 c 0.000 059 99 d 56 e 7.9 50 9 f 5 9 8.099 5 a D b C c B 6 a B b D cc d E 7 a nz7 b n C - 5 n 8 777 a7 6 8 a r 6 - s 6 b nt l +? nt 5 n 5 + itm c d p 9 ' - 0 6" x= a The power is reduced by. b Each answer is divided by to get the next answer. If the pattern continues we will get - = -= - =, etc. which illustrates a' = / an 9 ±,5 0=] =0 = =00 5=0 6=0 7= 8=000 9=00 0=00 Exercise Negative indices a b c x Y a5 x d e f 5a Y mn 6a g bc5 j ab a b 6 6 ab X y h a6 7 b '7a a i nia bn5 ab5 ft8 e m Y f 5Y x 6x m / a 5m0.7g 9 n 8p 6 b 6 a g y h x5 7 k g 8a6b6 agb
&fut - ARS k.i/_\qt-s cr-'0 -V- (0.) (c--) Ge) 900 0 ' 000 (:).0.,5 - Oco QS (6") coq 6s coo c:-.0c, Oovi I o no, Q-), CO cri 0-000 Zze, - 7 c o coo coo xtos xkot+ 0,-) CI) (e) cb) (a) 50 o cool-a'? 000 ocvo. 56 >c k0 8'0S <0 C0 to' (s-) x- (0-- () 5 -()(to7.) (-9 ocis - ic, x0-5 C) xko - f >c- (0- (k) -05 ) to - I. Co) Cio) (c-) (a) (e) X 0 - t ur fri -5 Ix to / c). C - I - to 5 ><Lo l G