Headstart Module 1: Indices

Transcription

1 206 Year 0 Core Mathematics Headstart Module : Indices Name: This work (Set - 4) must be completed for the start of next year, to a point as instructs by your teacher. It will be checked by the teacher on return to school and the whole topic will be assessed during the second week of term one 206. The assessment will be a SAC in the form of a 40 minute test. Contents: Set Evaluating Indices Set 2 First Four Index Laws Set 3 Scientific Notation ( Re: Scientific notation maths song media file) Set 4 Review. Media Files: Animation from Maths Online and SIMON If need be you can view instructional videos from the Headstart task via Maths Online or media files downloadable from SIMON. Lessons offered are: Multiplication with Powers (Indices) Division with Powers (Indices) Zero Power (Zero Index) Mixed examples with Powers (Indices) Negative Powers Changing Numerals to Scientific Notation Changing Scientific Notation to Numerals Assessment: Calculator Free SAC Set ( 4) inclusive

2 We often deal with numbers that are repeatedly multiplied together. Mathematicians use indices or exponents to represent such products more easily. For example, the expression 3x3x3x3 can be represented as 34. Indices have many applications in finance, engineering, physics, astronomy, biology, electronics, and computer science. 34 power or _ index or exponent base 3> 3 x 3 x 3, we can write 3 4. The 3 is called the base of the expression. The 4 is called the power or index or exponent If n is a positive integer, then an is the product of I/ factors of a. an=axaxaxax...xa n factors Find the integer equal to: a 2 5 a 25 =2x2x2x2x2 =32 Set Evaluating Indices ) Express in index form 2) a) x.x.x b) TY-Y-TY, c) 7. d) (-4)(-4)(-4) e) (8m)(8m)(8m)(8m) f) Copy, Expand and evaluate these common powers. Try to remember them. V b) c) 2 =..., 22=..., 2 3=..., 2 4= 4 = 28= 33=..., 34=, 4 2 =, 43=.., 44=, 2 5=,, 3 5=, 2 6 =, 3) Write in simple index form a) 8 b) 32 c) 27 d) 25 e) 0,000 f) 0,000,000 d) e) g) h) i) 5 = 6 = 9 =, 5 2 = 5 3 =, 5 4=, 6 2=., 6 3=., 6 4 = 83=..., 84=, 92=..., 93=...,,, 0 6=..., 06=...,

3 NEGATIVE BASES So far we have only considered positive bases raised to a power. However, the base can also be negative. To indicate this we need to use brackets. Notice that (-2) 2 = -2 x -2 whereas -2 2 = -(2 2) =4 = - x 2 x 2 = -4 Consider the statements below: (-) = - (-) 2 = - x - = (-) 3 = - x - x - = - (-) 4 = - x - x - x - = From the pattern above it can be seen that: (-2) 2 (-2) 3 (-2) 4 (-2) = = = = -2-2 x -2 x -2 x a negative base raised to an odd power is negative a negative base raised to an even power is positive. -2 = 4-2 x -2 = -8-2 x -2 x -2 = 6 Evaluate: a (_ 3)4 b -34 c (-3) 5 d -(-3) 5 Notice the effect of the brackets. a (-3) 4 =8 b x 3 4 = -8 (-3) 5 = -243 d = - x (-3) 5 = - x -243 = Simplify: a e -24 i -(-5) 3 c - 5 d -(-) 5 g -(-2) 4 h -(-5) 2 k (-3)4 I 72 5 Use your calculator to find the value of the following, recording the entire display: a 2 2 b (-5) 7 c -34 d 7 7 e 85 f (-9) h.22 I (_.08)23 Answers ) a) b), c) d) e) f) X3 5 Y (8)4 26 2) a) b) c) d) e) f) g) h) i) 2, 4, 8, 6, 32, 64, 28, 3, 9, 27, 8, 405 4, 6, 64, 256 5, 25, 25, 625 6, 36, 26, 296 7, 49, 336, , 64, 52, , 8, 729 0, 00, 000, 0000, , , 3) a) b) c) d) e) f) al C - b - 5 d l a 4096 c -8 b -'T82 d g23543 e -6 f 6 e f -729 g -6 It -25 g -729 I 25 JI - 27 k 8 I -49

4 In the previous chapter we saw that a x a x a= a 3. Using the same principle, the product a 3 x a4 can be written as (axaxa)x(axaxaxa)=a 7. Consider some other expressions and their simplifications: a 5 axaxax-a x a a 2 =a3 (axb) 2 =axbxaxb =axaxbxb = a2 b2 / 3 \ 2 k ) = X a X x axaxa a 6 3 a a a ( ) b b b b a 3 = b3 These examples can be generalised to the following index laws: axaxa bxbxb If the bases a and b are both positive and the indices m. and n are integers then: am x an = a -En a ril = a m n = a' (ab) = a" bm (a n an b) b" a =, a 0 0 a rt an Example Simplify (a),c4 x x 6 To multiply numbers with the same base, keep the base and add the indices. To divide numbers with the same base, keep the base and subtract the indices. When raising a power to a power, keep the base and multiply the indices. The power of a product is the product of the powers. The power of a quotient is the quotient of the powers. Any non-zero number raised to the power of zero is. and in particular, a- = ā. (b) x4y4 x x 6 y" (c) 2a 6 x 5a 7 Solution (a) x4 x x6 = x4+6 = S i (a) x4 x x6 = x4+6 = x 0 (b) x4y4 x )0y' = x4+6 x y4+7 (c) 2a 6 x 5a 7 = 0a6+7 = xioyi = 0a 3 Note: multiply the numerical coefficients but add the indices.

5 Example 2. Simplify (a) x6 x 4 (b) x 6y 0 x4y2 (c) 8x 2x 4 (d) 3x" ± 8x 4 Solution (a) x 6 x4 = x 6-4 (b) x6y to ± x4,, y x y = X 2 = X 2 y 8 (c) 8x" ± 2x 4 = 8x" = 4x 6 2x 4 x 0-4 = 4 (d) 3x 2 Note: divide the numerical coefficients but subtract the indices. 8x 4 = 3x2 8x4 = -x = - 6x8 Example 3 Simplify (a) (X4 ) 2 (b) (2x 2 )3 Solution (a) (x 4 ) 2 = X 4 x 2 Example 4 Evaluate (a) 5 (c) 2a Solution (a) 5 = = X 8 (b) (2x 2 )3 = 23x2 x 3 = 2 3 x 6 = 8x6 (b) (2a) (b) (20 = y4) 3 x4 x 3 (C) = 2 x 3 X 2 =.r (c) 2a = 2 x Set 2 Index Laws. Simplify (a) a 6 x a' x (c) b x b2 x b4 (e) a 2 b 3 x b 7 (g) abc 2 x ab2 c x a 2 bc (i) 3 a2 b4 x 6a 2 b 3 (k) 2 2 rn 4 n 5 32 m 6 n (m) 4x4y5 x 4 2 x 3 y 7 (o) 2a4 b 7 x 3a 2b x 3 2 ab4 (b) x4 x x 3 x x 2 4 x m" x m (d) n2 (f) x 5 y6 z 3 x xy 2 z (h) mn4 p2 x m3 n4 p2 8a4 5,6 x 9a3 () 5 2 m 2 n4 x 2 2 m 5 n 7 (n) 3 2 n4n2 x 3 2 nin (p) 3 5)6' x 3xy x 32x2y

6 2. Simplify a 6 (a) a4 X 9 (C) x3, 9x 2 te) 3x (g) 0x 3 20x 3 5x 6 (i) 25x 3 5a2b3c5 k) ( 2ab 2 c 3 (m) 26 a4b 3 c8 24a 3 b 7 c 5 3. Simplify. (a) (a 4 ) 3 (2x4 )(C) 2 2 y3 (e) (4x )4 c)2 (a) y (k) ( (3X4 ) 3 y 2 ( 9ac 23b3 ) 4 4. Evaluate (a) a (c) 3a (e) (4a) (g) (5a2 b3 ) (i) 3a 3 (k) 5a + 4 (b) (d) a 0 a X 20 24x 7 (f) 3x 3 25x6 (h) 00x6 6a a () (n) 27a 7 b 5 c4 6a6 b 3 c4 34a7b5 cio 34ab4c5 (b) (m 3 )7 (d) (9x 3 )2 (0 (3m 2 n 2 p 3 )3, (h) 2 u/ (4m2)5 n 2 () (7m2n3) 2 \P (b) c (d) 5m (f) (3x) (h) (6a 6b 3 ) 5a + 7 (j) () 7a 7. (a) a' 6 (b)./c9 (c) b7 (d) m9 (e) a8ble (0 x6y8z4 (g) a4 b4 c4 (h) m4n8p4 (i) 8a4b7 (j) 72a 7bi 3 (k) 36m"n 6 (I) 00m n" (m) 64x 7y 2 (n) 8msn 7 (o) 54a 7b" (p) 3 8x 4y 4 2. (a) a 2 (b) a 3 (c) x 6 (d) x' (e) 3x (f) 8x 4 (g) (h) (i) X 3 (j) -,sra 2 (k) iabe 2 (I) lab 2 c (m) 4ab 6 c 3 (n) Obc 5 V x". 4 5m" 3. (a) a'2 (b) m 2 (c) 4x 8 (d) 8)C 6 (e) 44 x 8y' 2 (f) 27m 6n 6p 9 (g) (h).0) (j) y 2 z2 Y6 n t0 94a 8b 2 49m 4n 6 (k) (I) 4. (a) (b) (c) 3 (d) 5 (e) (f) (g) I (h) (i) 0 (j) 2 (k) 9 (I) 0

7 Negative Indices Using the 2nd law of indices we see that 2 = 2' and 2 =, thus = 3 o Similarly 32 = 3-2 and 3o =, thus 32 = 3-2 In general = n # 0 a" It can be shown that the laws of indices are obeyed for negative indices. Terms are considered to be in their simplest form when expressed with positive indices. Example Express each of the following with a positive index: (a) 4a - 3 (b) 3-2 b 7 )) 3-2 b - (c) 4 x -2 (d) x 2 y -3 x x -4 3/ 5 Solution (a) 4a -3 = 4 a 3 x 7 y (e) _ x y (only the a is raised to the power of 3) (f) (3x 2 )-2 (b) 3 2 b 7 = 3 2 b 7 (c) 4 4 = T 2 X = 4 x = 4x 2 x 2 (d) x -2 y-3 x x -4y 5 = x -6 y 2 y 2 = X X 7 y -2 (e) -4 2 = x y x = Y4 (4x 2V x -6 (g)( ) Y -3 Recommended procedure for simplifying index expressions (i) remove brackets (ii) apply index laws (iii) express with positive indices Y X 6

8 Set 3 Negative Indices. Express with positive indices (a) a - 3 (C) (e) (g) 5 (h) (i) 3a -4 (j) 5x -7 (k) 4a -5 () 9m (m) (n) a4 x - 5 (0) 3-2 a - 2 (p) 4-2x -2 (q) 7-3 a -5 (r) 5-5 x Simplify, expressing your answer with positive indices (a) x 7 6 y4 x x 2 y- 2 (b) a -3 b -5 x a 5 b -3 (c) 3x -2 y 5 x 5x -7 y -2 (d) 2a-l b 5 x 7ab -3 (e) 7a3 m -4 x 8a 5m' () x 4r -3 s 5 8a -4 6a -4 (g) (h) 2a 6 8a 5 ( i) 44x7y5 2x - 3 3,4 7a2 b - 3c -4 (k) 2a 5 b -7 c -9 (m) (3x) -2 (o) ( (q) (3a 2 b -2 ) 3 x (2a4 ) a4b- 3 (i) 36ab 2 9m 3 n4p - 5 () 2m - 3 Yl4p 2 (n) (4y) - 3 (3) (5 2x 3 ) - 5 (r) (5x4y6 ) - 3 x (5 2xy - ) (a) (b) (c) (d) (e) x 3 (f) x' (g) y 4 (h) a' 0) (k) (I) (m) 3a 4 a 3 x 7 m 5 n 4 a4 x 7 a' m 7,2 a 2 5,3 I 56 (n) 5x 5 (o) (p) (q) (r) 2. (a) ' (b) (c) ' (d) 46 2 (e) (f) 9a 2 6x 2 73a5 557 a2m7 x 4 ba X (g) - 4j- (h) --3- (i) 2x"y (j) (k) b4c5 0) 3m0 (m) 9x, (n) - (0) _. 4 - _.. (p).-;-i (q) 4aio (2 0 a9 b 3a 3 7p 7 'y' 2'"y" 25 x x 8 2a" a' 8a 4 6c" (r) (s) (t) ( ) (N) mn 3P9 (w) (x) (Y) (z) x 9y 2 ' y 7 3y b 5 c 3 b 7 b 6 d5 9m3n9p

9 There are many situations where we need to describe very large and very small numbers. For example: There are about cells in the human body. A glass of tap water contains about grams of chlorine. Numbers with so many digits can be hard to comprehend and operate with. We can use scientific notation to write these numbers in a way that is easier to understand. Scientific notation involves writing any given number as a number between I inclusive and 0, multiplied by a power of 0. The result has the form a x 0k where a < 0 and k is an integer. If the original number is greater than or equal to 0, then k is positive. If the original number is less than, then k is negative. If the original number is between and 0, we write the number as it is and multiply it by 0, which is. To write a number larger than 0, we start with a number between and 0, then multiply it by a positive power of 0. For example: 27 = 2.7 x =- 5.8 x = 3.04 x 000 = 5.8 x 02 = 3.04 x 03 We can express small numbers in the same way. We start with a number between and 0, then then divide it by a power of 0. This is the same as multiplying by a negative power of 0. This is the same as multiplying by a negative power of 0. Set 4 Scientific Notation For example: = = = 7.5 x = u A = 7.5 x 0- = 6 x 0-4 The following values are all equal to Which of them is written in scientific notation? A 53 x 03 B 0.53 x 0 5 C 5.3 x 04 D x 00 2 Copy and complete to write the following numbers in scientific notation: a x W"- b 8000 = x 0 3 c 0.04 = x 0-2 d = 5.07 x 0-- e x 06 f = 2.3 x 0--

10 Express in scientific notation: a b a = 4.5 x = = 5.92 x 04 = 5.92 x Express in scientific notation: a 425 b c 4.25 d e f h I k I 0 4 Express in scientific notation: a The distance from the Sun to Mars is about km. b A set of kitchen scales are accurate to within kg. c A bacterium is smaller than mm. d The probability that your six numbers will be selected for Lotto on Monday night is e The central temperature of the Sun is 5 million degrees Celsius. f The Great Barrier Reef covers an area of hectares. Write as an ordinary decimal number: a 3.2 x 02 b 5.76 x 0-6 a 3.2 x 02 = 3.20x 00 = 320 b 5.76 x 0-5 = = Write as an ordinary decimal number: a 5 x 04 b 3 x 0 3 C.8x io 7 d 8. x 02 e 6.5 x 05 f. x 0 g 2.75 x 08 h 8 x 06 6 Write as an ordinary decimal number: a 3 x 0-2 b 9 x 0-5 c 7 x 0-3 d 4. x 0-4 e 8.2 x 0-6 f 7.6 x 0 - g 3.25 x 0-7 h 2 x 0-8

11 7 Express as ordinary decimal numbers: a The wavelength of blue light is 4.75 x 0-7 m. b The estimated world population for 200 was 6.85 x 0 9 people. c Physicists in Japan created a model bull which is only.2 x 0-5 m long. d The length of the Earth's equator is approximately 4.0 x 0 4 km. e A mosquito weighs about.5 x 0-6 kg. Simplify the following, giving your answer in scientific notation: a (5 x 04) x (4 x 0 5 ) b (8 x 05) (2 x 03) = 2 x 0 a (5 x 04) x (4 x 0 5 ) b (8 x 05) (2 x 03 ) = 5 x 4 x 04 x x 0 5 = 20 x x 0 3 = 2 x 0 x 0 x = 4 x 02 8 Simplify the following, giving your answer in scientific notation: a c e g (2 x 0 5) x (3 x 0 2) (8 x 05 ) x (5 x 0 6) (3 x 04 ) 2 (8 x 02) (2 x 0 3) b d f h (4 x 03) x (6 x 0 3 ) (9 x 09 ) 2 (7 x 0-3 ) 2 (6 x 07) (3x 0 4) SCIENTIFIC NOTATION ON A CALCULATOR Calculators use scientific notation to display very large and very small numbers. However, if we wish we can tell the calculator to give all its answers in scientific notation. Instructions for writing numbers in scientific notation can be found by clicking on the icon. Display readout Av The &splay comprises the entry line, the result line, and indicators. Math!' A Indicator Entry line A x75 32)( xio3 Result line C'ALCU LATOR - INSTRUCTIONS [Math] : 2.75 x 0 5 = 2.75 x [xl0x][( )]5 [= [ FOID] MathI x o - 5 Maths 2.75x x O's

12 )0 [Line] : 2.75 x 0-8 = 25 x 0-8 [SETUP ] [ 2 (Line0) 2.75 [x0r][(-)]5 [SET ] [ 8 ][ 2 ] (NORM 2) A x i o x 0' x o [Line] 0000 x 0000 x 00 = 0,000,000,000 = x [ x ] 0000 [xi 00 [= 0000 x 0000 x lx 0 g Calculate the following, giving each answer in scientific notation. The decimal part should be written correct to 2 decimal places. a x 0.0 ± 5000 b 375 x 220 x c x d 800 x 740 x e ± f 0.09 x x Find in scientific notation, with the decimal part correct to 2 decimal places: a (2.8 x 08) x (3.4 x 0 4 ) b (9.8 x 0-4 ) x x x 06 d (8.66 x 0-3) x (7.5 x 0-8 ) f (7.59 x 04 ) 3 For questions and 2 give your answers in scientific notation, with the decimal part correct to 2 decimal places. Assume that year days. Australia's population is approximately 22 million people. Each Australian uses an average of 220 litres of water per day. Estimate the total water consumption by Australians: a in one day b in one week c in one year. 2 Light travels at a speed of x 0 8 metres per second. How far will light travel in: a minute b day c year? Answers C 2 a 376 = 3.76 x 0 2 b 8000 = 8 x 03 c 0.04 = 4 x 0-2 d = 5.07 x 0-3 e = 9,04x 0 6 f = 23x a 4,25x 02 b 4.25 x 05 a 4.25 x 0 d 4,25)< 0- e 4.25 x 0-5 f 2.0 x 0 g 2.0 x 04 h 2.0 x 0-3 I 2.0 x 06 J 2.0 x 09 k 2.0 x 0-6 I x 0 4 a 2.49 x 05 km b 5 x 0-4 kg a 5 x 0-3 mm d.462 x 0-7 a.5 x 07 C f x 07 hectares 5 a b 3000 c d f g h e f g h a 0.03 b c d f 0.76 g h a in b people c m d km e kg 8 a 6 x 07 b 2.4 x 07 c 4 x 0 2 d 8. x 0 9 e 9 x x 0-5 g 4 x 0 - Ii 2 x 03 9 a 6.00 x 0- b 2.39 x 09 c 2.9 x 03 d 4.0 x 00 e 6.24 x 0-9 f.03 x a 9.55 x 09 b 9.62 x 0-7 c 4.90 x 0-4 d 6.50 x 0-7 e 4.00 x 0-7 f 4.37 x 04 a 4.84 x 09 litres e.77 x 02 litres b 3.39 x 09 litres 2 a.80 x 00 m b 2.59 x 03 m e 9.46 x 05 m

13 Set 4 Review A. Evaluate each of the following: (a) 2 3 (b) 3 3 (c) 4 - ' (d) 2-3 (e) 5-2 (f) 3-i 2. Simplify each of the following expressing the answer with positive indices. (a) a 2 x a4 x a 6 (b) m 3 x m 5 x m (c) p 3 x r x p 4 x r 2 (d) 2t 5 x 6s3t4 x s - I t -3 (e) 5m -3n - 2 X 6m 4n - i (g) a -2 b -3 x a-3b-3 (f) 5p -6q - 2 X 2p3q4 x py (h) f -4g 2 h - 7 x 3f2g - h5 (i) M 5 n4 () 4a 6 b 3 i m 2 n 3 2a 3 b 2 2p2q 3r 5 32m 3n2 (k) 3p4q4r3 () then' 6C 3d 2 Set 5 Review 5p2g3 (m) (n) 4c 2d - 0p 2q - 2 2s -2t - 5 4f - 2g 3h - (0) (p) 0S - 3t4 2f 3g - 2h2 (q) (2a 3 b4 ) 2 (r) (5m -2n - 3 ) 3 (s) (S 3t - 4 ) - 3 () (3r 3 V - 6) - 2,, S 3t 3) - 2 (pi.. 2g2)- 2 () (- (v) r2 r - 3 (a2b3)- cre n5)- 3 (w) (X) 2 c-2 P - 3. Simplify each of the following expressing the answer with positive indices. (a) (a 3 b 2 ) 2 x (a- 2 b - 3 ) - 2 (b) (m -2 n 3 ) - 2 x (m 3n -4)- 30m P'. (a) 8 (b) 27 (c) / (d) -4- (e) (f) i 2. (a) a' 2 (b) m 9 (c) (d) 2s 2 t 6 (e) (f) 60pq" r n 3 3g 2 2g 5 (g) (h) (i) 'On (j) 2a 3b (k) L r2 s (I) (m) 4cd 3 (a) 4q 5 (o) (p). (q) 4a 6b" p 2 q mn3 a s h f 2 h2 5t 9 3f 5 h r4 n4 m (r) (s) (t) 02 (u) (v) ' (w) (x) 3. (a) a' b' (b) (c) 4a m o no s 9r6,6 6 q4,6 a 2 b 3 c 2 27p 6m 2n" n 2 b 6 (d) 27h" (e) (0 3a2 m3n8 2b3 m4n2 (g) p 0, 0 (h). P"

14 B Set 4 25 Write as a power of Write as a power of Write as a power of Write as powers of 2 or 3: a 4 29 Write as powers of 2 or 3: a 8 30 Write as powers of 2, 3 or 5: a 32 c27 3 Express the following in standard form (scientific notation): a 365 b 289 c Express the following in standard form (scientific notation): a 42.6 b c Express the following in standard form (scientific notation): a b c The River Nile is approximately m long. Express this value in standard form (scientific notation). 35 One particular species of wasp has a body length of cm. Express this value in standard form (scientific notation). 36 The planet Saturn is km from the Sun. Express this value in standard form (scientific notation). 37 Write as an ordinary decimal number: a 6 x 02 b 2.8x 0-3 c 65x 0-38 Write as an ordinary decimal number: a 6.2 x 03 b 6.8 x 04 c 3.2 x Write as an ordinary decimal number: a 3.09 x 03 b.284 x 06 c x An Olympic swimming pool contains approximately 2.5 x 06 L of water. Write this value as an ordinary decimal number. 4 An average sized snow crystal has a mass of around 2.9 x 0-8 kg. Write this value as an ordinary decimal number. 42 The average distance from Earth to the Moon is 3.8 x 0 3 km. Write this value as an ordinary decimal number. 43 Simplify the following, giving your answer in standard form: a (2 x 03 ) x (4 x 02 ) b (6 x 02 )2 44 Simplify the following, giving your answer in standard form: a (6 x 05 ) (2 x 02 ) ti (7 x 0-2 )2 45 Simplify the following, giving your answer in standard form: a (.2 x 08 )2 b (3x 04) (4 x 0-3 ) 46 Write each of the following as it would appear on the display of a calculator in scientific notation: a b

15 a 47 Write each of the following as it would appear on the display of a calculator in scientific notation: a 7.29 x 07 b Write each of the following as it would appear on the display of a calculator in scientific notation: a b 4.35 x Use your calculator to write the answer to each of the following in scientific notation, correct to 2 decimal places: a 0.08 x b 0 x 500 x Use your calculator to write the answer to each of the following in scientific notation, correct to 2 decimal places: a 650 x 2000 x b Use your calculator to write the answer to each the following in scientific notation, correct to 2 decimal places: a b 0.98 x x Use your calculator to write the answer to each of the following in scientific notation, correct to 2 decimal places: a (7.5 x 03) x (8.6 x 05 ) b (.5 x 0- )2 53 Use your calculator to write the answer to each of the following in scientific notation, correct to 2 decimal places: a (5.2 x 08) x (3.9 x 05 ) b (6.57 x 04 ) (2.28 x 0-6 ) 54 Use your calculator to write the answer to each of the following in scientific notation, correct to 2 decimal places: a (4.9 x 07 ) (2.5 x 0-4) 6.94 x 02 Answers = = = 0-6 a 4 C 27 = 2 x 2 = 3 x 3 x 3 = 22 _ 3 3 a 8 = 2 x 2 x 2 = 23 a 32 =2 x 2x 2 x 2x 2 = = 3.65 x 00 = 3.65 x = = 6.5 x 0 b rt85 =.289 x =.289 x a 33 a c.6.6 = 4.26 x 0 = 4.26)< 0 c8 =.9468 x =.9468 x 05 = x 000 = x = = 3.24 x in = 6.7 x in = 6.7 x 06 in 0.R39 cm = cm =.39 x 0-3 cm 33676:5366 km =.35 x km =.35 x 09 km b b 0..5FRC2 = 8.24, 04 = 8.2 x 0-4 = x 07

16 37 38 a 6 x 02 = 6 x 00 = 600 c 65 x 0- r5 0 = 6 = 6.5 a 6.2 x 03 = 6,2>< 000 = 620 c 3.2 x 0-3 = = b b 2.8 x 0-3 -= r = x 04 = 6.700a x = a (.2 x -08 )2 = (.2 x 08 ) x (.2 x 08) =.2 x.2 x 08 x 08 =.44 x =.44 x 0 b (3 x 04 ) (4 x 0-5 ) 3 x 04 4 x 0-5 = x 4 = 4 X = 0.75 = 7.5 Ur x 09 x 39 a 3.09 x 0 = 3:0570 x 05 = c x 0-5 = f0i5( = b.284 x 06 =.i00x 06 = a 54'760 = 5.4 = 54 i.e., x 00 x b = = 6.42 x 0-3 i.e., x 06 L = 2:5750O x 06 L = L 49 a 0.08 x b 0 x x 6000 = 2 x 0-3 = 3.3 x x 0-8 kg = kg = kg 3.8 x 05 km = a ) x 05 km = km 50 5 a 650 x b 25 ± 2000 x =.56 x x 06 = 5 a ± = 2.3 b 0.98 x x 0.06 = x 0-4 *4.8-4 x 0(2 dee pl) a (2 x 03 ) x (4 x 02 ) = 2 x 4 x 03 x 02 = 8 x 03+2 = 8 x 05 a (6 x 05 ) (2 x 02 ) 6 x 05 2 x 02 - x = 3 x 03 b (6 x 02 ) 2 = (6 x 02 ) x (6 x 02) =6 x 6x 02 x 02 = 36 x 02+2 = 36 x 04 = 3.6 x a (7.5 x 03 ) =_ 6.45 x 09 b (.5 x a 0- )2 = 2.25 x 0-2 (5.2x 08 ) = x 04 * x 0 x (8.6 x 05 ) x 05 ) x (3.9 (2 dec pl) b (6.57 x 04 ) (2.28 x 0-6 ) = x 00 *2.88 x 0 (2 dec pl) b (7 x 0-2 )2 = (7 x 0-2 ) x (7 x 0-2) = 7 x 7 x 0-2 x 0-2 = 49 x = 49 x 0-3 =4.9 x 0 54 a (4.9 x 07) (2.5 x r 4 ) *.95 x 0 (2 dec pl) 6.94 x 02 ±.44 x 0-3 (2 dee pl)