INDICES WHAT S IN CHAPTER 7? IN THIS CHAPTER YOU WILL:

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1 INDICES 7 IN THIS CHAPTER YOU WILL: WHAT S IN CHAPTER 7? 7 0 Multiplying terms with the same base 7 0 Dividing terms with the same base 7 0 Power of a power 7 0 Zero and negative indices 7 0 Index laws review 7 0 Significant figures 7 07 Scientific notation for large numbers 7 08 Scientific notation for small numbers 7 09 Scientific notation on a calculator multiply and divide terms with the same base find a power of a power use zero and negative indices round numbers to significant figures understand and use scientific notation for large and small numbers use a calculator to evaluate expressions with scientific notation Shutterstock.com/Sdubi ISBN Chapter 7 Indices

2 7 0 Multiplying terms with the same base WORDBANK base The main number that is raised to a power, for example, in the base is. power The number at the top right corner of the base that represents repeated multiplication by itself, for example, means and the power is. index Another word for power. indices The plural of index, pronounced in-da-sees. When multiplying terms with the same base, such as, we can add powers because: = ( ) ( ) = 8 ( + = 8) times times To multiply terms with the same base, add the indices. a m a n = a m + n EXAMPLE Simplify each expression, writing the answer in index notation. a b x x c w w w d m n a = 7 b x x = x c w w w = w 7 d m n = 9m n Add the indices if the bases are the same w = w The bases are different here. EXAMPLE Simplify each expression. a a a b m ( m ) c 7e f ( e f ) a a a = a a b m ( m ) = ( ) m m = a = 8m c 7e f ( e f ) = 7 ( ) e e f f = 8e f Developmental Mathematics Book ISBN istockphoto/richcarey

3 exercise 7 0 Simplify. Select the correct answer A, B, C or D. A 8 B 8 C 9 D 9 Simplify. Select A, B, C or D. A 8 B 9 C D 9 8 Copy and complete each statement. a means c The index of is b The base of is Write each expression using index notation (powers). a b c 7 7 d e x x x x x f a a a g h w v v v i a a j m n m n n k u u v l m n Simplify each expression using index notation. a b c 7 d 0 e m m f x x g p p h n n Copy and complete each solution. a m m = m b w w 7 = ( ) w = m = w 7 Simplify each expression. a n n b a 8a c n n d w 8w e v v f n ( n ) g a ( 9a ) h a ( a ) i c c 8c j a b 8a b k m n 8m n l a c ( 8a c ) 8 Is each statement true or false? a = 8 b = 0 c = d = 9 Can be simplified by adding indices? Use your calculator to evaluate it. 0 Use a calculator to evaluate each expression. a b c d 7 e f 7 g ( ) h ( ) ISBN Chapter 7 Indices 7

4 7 0 Dividing terms with the same base When dividing terms with the same base, such as, we can subtract powers because: = = = ( = ) To divide terms with the same base, subtract the indices. a m a n = a a m n = am n EXAMPLE Simplify each expression, writing the answer in index notation. a 8 b m 0 m c 7 7 x x a 8 = b m 0 m = m c 7 7 x 7 = 7 d = x x Subtract the indices if the bases are the same x = x EXAMPLE Simplify each expression. a w w b 8x ( 7x 0 ) c mn 9mn Divide coefficients first. a w w = w b 8x ( 7x 0 ) = 8 w 0 7 mn c = x 9 9 m n mn m n = w 0 = x = m n Shutterstock.com/AnetaPics 8 Developmental Mathematics Book ISBN

5 exercise 7 0 Simplify 8. Select the correct answer A, B, C or D. A B C D Simplify 8. Select A, B, C or D. A B C D Copy and complete each solution. 8 8 a = b x x = x = = x c = = 9 Simplify each expression. a b 8 c d 0 m e d 8 d f m g x x h m 8 m Simplify each expression. a n 8 n b 8a a c n n d w 8w e 8v 8 v f n 9 ( n ) g a 7 ( 9a ) h a ( a ) i c 8c j a 8 b 8a b k m 7 n m n l a 8 c ( a c) Is each statement true or false? a = b x x = x 8 7 m 8 c = 7 d = m 7 m 7 Can be simplified by subtracting indices? Use your calculator to evaluate it as a fraction. 8 Use a calculator to evaluate each expression correct to decimal places where appropriate. a 8 b 9 8 c d 8 ISBN Chapter 7 Indices 9

6 7 0 Power of a power When finding a power of a power with the same base, such as ( ), we can multiply powers because: ( ) = = ( ) ( ) = times times To find a power of a power, multiply the indices. (a m ) n = a m n = a mn EXAMPLE Simplify each expression, giving each answer in index notation. a ( ) b (n ) c (x ) 0 a ( ) = b (n ) = n c (x ) 0 = x 0 = 0 = n 0 = x 0 Multiply the indices To find a power of ab or a, raise a and b to the power separately. b (ab) n = a n b n and a b n = a b n n EXAMPLE Simplify each expression. a (x ) b m c a b Always raise each part in the bracket to the power separately. a (x ) = (x ) b m = ( = x = m c a b = a ( b ) = 9 a 8 b Shutterstock.com/Pakhnyushcha 0 Developmental Mathematics Book ISBN

7 exercise 7 0 Simplify ( ). Select the correct answer A, B, C or D. A B 8 C D Simplify (a ). Select A, B, C or D. A a B 9a C 9a D a Copy and complete. a ( ) = = b (x ) = x _ = Simplify each expression, giving each answer in index notation. a ( ) b ( ) c ( ) d ( ) e (7 8 ) f (x ) g (n ) h (m 8 ) i (w ) j (a ) k (9 ) l (x ) m ( ) n (q ) o (p 8 ) 7 Is each statement true or false? a (a ) = a b (x ) = 7x c (n ) = n 8 Simplify each expression. a (x ) b (a ) c (n ) d (m 8 ) e (7c ) f (w 8 ) g (b ) h (9t ) i ( a ) j (w 8 ) k ( c 7 ) l ( q 9 ) 7 Is each statement true or false? x x a = 7 b 8 w = c w a = a 8 8 Simplify each expression. a a b m c w x 8 d m e f w 7 g 8 c h n 9 Is each statement true or false? 0 a a a = b a a b = b 0 Simplify each expression. 8 x a b m n c a d a b ISBN Chapter 7 Indices

8 7 0 Zero and negative indices What does equal? = = But also, when dividing terms with the same base, we subtract indices: = = 0 So 0 =. What does equal? = = But also, when dividing terms with the same base, we subtract indices: = = So =. Any term raised to the power of 0 is. a 0 = A term raised to a negative power gives a fraction with a numerator of and a denominator that is the same term raised to a positive power. a n = a n EXAMPLE 7 Simplify each expression. a 0 b a 0 c (n) 0 d ( w) 0 a 0 = b a 0 = c (n) 0 = d ( w) 0 = = EXAMPLE 8 Simplify each expression. a b a c x d m a = b a = a = 9 c x = d x = x m = m = m istockphoto/tpuerzer Developmental Mathematics Book ISBN

9 exercise 7 0 Write a with a positive index. Select the correct answer A, B, C or D. A a B a C 8 a D 8 a Write with a negative index. Select A, B, C or D. x A 9x B x C x D 9x Is each statement true or false? a 0 = b 7 0 = 7 c n 0 = d (n) 0 = Simplify each expression. a 0 b 0 c 0 0 d x 0 e m 0 f y 0 g 0 h x 0 i (a) 0 j m 0 k ( n) 0 l a Copy and complete each statement. a = b w = w n = n Write each term with a positive index. a b c d e f 7 7 Evaluate each expression. a 0 b c d e 9 0 f 7 8 True or false? a = b = c = d 0 = 0 e 0 = f 8 = 0 9 Simplify each expression. a x b a c m 7 d w e x f n g x h a i u j w 0 Evaluate each expression. a b c d e 0 ISBN Chapter 7 Indices

10 7 0 Index laws review This table summarises all the index laws learnt so far. Note that these laws only apply to expressions in which all terms have the same base. When multiplying, add indices When dividing, subtract indices To find a power of a power, multiply indices To raise ab to a power: To raise a to a power: a b a m a n = a m + n a m a n = a m n (a m ) n = a m n (ab) n = a n b n b n = a b n n To find a zero index: a 0 = n To find a negative index: a = a n EXAMPLE 9 Simplify each expression. a a b ab b mn ( mn ) c (a ) (a) 0 a a b ab = a a b b b mn ( mn ) = = a b 9 m n m n = n 8 = n 8 c (a ) (a) 0 = a = 8a Shutterstock.com/limor.com Developmental Mathematics Book ISBN

11 exercise 7 0 When dividing terms with the same base, the indices are what? Select the correct answer A, B, C or D. A added B subtracted C multiplied D divided When finding a power of a power, the indices are what? Select A, B, C or D. A added B subtracted C multiplied D divided Is each statement true or false? a a a = 8a 0 b 8m m = m c x 0 = d (n ) = 9n Simplify each expression. a b c 7 7 d 0 e w w f c 7 c g x x h a a 0 Simplify each expression. a b 7 7 c d c e x x f c g a w w h a Simplify each expression. a ( ) b ( ) c ( ) d ( ) e ( ) f ( ) g ( ) h ( ) 0 i (x ) j (a ) k (w ) l (p 8 ) m (a ) n (x 7 ) o (w 0 ) p (x ) 7 Simplify each expression. a x x b a a c (m ) d a a m e m f (a ) 8 w g (x ) h 8w i 8a b 7 ( a 0 b ) j m 8 n m n k w 0 ( w ) l x y ( x y ) 8 Is each statement true or false? a a 8 a = a b 8m m = m c (x ) = x d (a ) = 8a m e w w = 8w f = m m g (x ) 0 = h a x = 8a ISBN Chapter 7 Indices

12 7 0 Significant figures We already know how to round numbers to decimal places, but we can also round to significant figures. The significant figures in the number are,, 7 and because they indicate the size of the number. The 0s at the end are not significant. The significant figures in the number are 9, 0,, and, but the two 0s at the end are not significant. The significant figures in the decimal 0.08 are, and 8 because they indicate the size of the decimal. The 0s at the start are not significant. So only 0s at the end of a whole number or at the start of a decimal are not significant. EXAMPLE 0 Round each number to significant figures. a 98 b 000 c a The rd digit is rounded up to 7 as the next digit 9 >. b The rd digit is rounded down as the next digit <. c The rd digit 7 is rounded up to 8 as the next digit is. EXAMPLE Round each number to significant figures. a b c. Count the first non-zero digits in each number. a The nd significant figure is rounded up to as the next digit 8 >. b The nd significant figure 0 is rounded down as the next digit <. c..7 The nd significant figure is rounded up to 7 as the next digit >. Developmental Mathematics Book ISBN Shutterstock.com/SARANS

13 exercise 7 0 Round 8 to significant figures. Select the correct answer A, B, C or D. A B 8 C 9 D Round to significant figures. Select A, B, C or D. A 0.00 B C 0.00 D Round each number to significant figures. a 8 b 8 8 c 8 0 d 8 e f 7 g 000 h 7 8 i j k 0 8 l Write each number correct to significant figures. a b c 0.8 d.00 e f 0.07 g h 0.70 i.800 j k.0 l At the Australian Open tennis in Melbourne, there were 8 7 people in Rod Laver Arena. Round this number correct to: a significant figure b significant figures c significant figures Shutterstock.com/Neal Cousland The concentration of salt in some beach water was Round this number correct to significant figures. 7 The population of Inandarra was 78 in the last census. a If the population was increasing at a rate of % per year, find the population in year s time. b Round this number correct to significant figures Calculate correct to significant figures ISBN Chapter 7 Indices 7

14 7 07 Scientific notation for large numbers WORDBANK scientific notation A way of writing very large or small numbers, using a decimal from up to but not including 0 multiplied by a power of 0, such as to write A number written in scientific notation has the form m 0 n, where m is a number between and 0 and n is an integer. To write a large number in scientific notation: use its significant figures to write a decimal between and 0 for the power of 0, count the number of places in the number after the first digit EXAMPLE Write each number in scientific notation. a 7 00 b c a 7 00 = 7. 0 places underlined after the first significant figure, 7 b = places underlined after the first significant figure, c = places underlined after the first significant figure, EXAMPLE Write each number in decimal form. a. 0 b c 0 9 a. 0 = 00 Move the decimal point places right or make places after the. b = Move the decimal point places right or make places after the 8. c 0 9 = Move the decimal point 9 places right. EXAMPLE Write these numbers in ascending order: To compare numbers in scientific notation, first compare the powers of 0. Both. 0 and.0 0 are definitely smaller than 0 7 because 0 < 0 7. Since. 0 and.0 0 have the same power of 0, we need to compare their decimal parts..0 0 is smaller because.0 <.. So in ascending order, the numbers are.0 0,. 0, 0 7. OR: Convert the numbers to decimal form to compare them.. 0 = = = Developmental Mathematics Book ISBN

15 exercise 7 07 Write in scientific notation. Select the correct answer A, B, C or D. A.8 0 B 8. 0 C 8 0 D.8 0 Write.0 0 in decimal form. Select A, B, C or D. A 0 00 B C 00 D 0 Copy and complete each statement. a 000 =. 0 b = 0 c 800 = 0 d = 0 e = 0 f = 0 Write each number in scientific notation. a 000 b 000 c d e f g h Copy and complete each statement. a 7. 0 = 7... b 8. 0 = 8... c. 0 8 =... d.0 0 = 0... e.98 0 = f.78 0 = 7... Write each number as a basic numeral. a 0 b.8 0 c.8 0 d. 0 7 e 0 f 9. 0 g h a Write the numbers below in ascending order b List these same numbers in descending order. 8 Write each number in scientific notation. a The dam has the capacity to hold 8 megalitres of water. b There were Australians attending the Anzac Day service in Gallipoli. c The population of Australia was approximately d The distance from Earth to the Sun is about kilometres. 9 Write your answers to each problem in scientific notation. a Chen enters a walkathon to raise money for charity. He walks km each day for 8 days. How far has he walked altogether? b Hannah flies to Rome in a plane travelling at 800 km every hour. The flight is non-stop and takes 8 hours. How far did she fly? c Liam is in training and swims km every day for 8 weeks. How many kilometres has he swum in this time? d Sophie is an author. She types 000 words every day. How many words will she type in a year? e Rajan delivers leaflets in mailboxes every day for 8 weeks. If he is able to deliver 0 leaflets per day, how many leaflets are delivered in 8 weeks? 0 On the Internet investigate how many kilobytes are in a gigabyte and write your answer in scientific notation. ISBN Chapter 7 Indices 9

16 7 08 Scientific notation for small numbers A number written in scientific notation has the form m 0 n where m is a number between and 0 and n is an integer. To write a decimal in scientific notation: use its significant figures to write a decimal between and 0 for the negative power of 0, count the number of places in the number up to and including the first significant digit (or count the number of 0s) EXAMPLE Write each number in scientific notation. a 0.00 b c a 0.00 =. 0 places underlined up to the first significant figure, (or 0s) b = 9 0 places underlined up to the first significant figure, 9 (or 0s) c =.0 0 places underlined up to the first significant figure, (or 0s) EXAMPLE Write each number in decimal form. a. 0 b c 9 0 a. 0 = b = Move the decimal point places left or insert 0s in front. Move the decimal point 7 places left or insert 7 0s in front. c 9 0 = Move the decimal point places left or insert 0s in front. EXAMPLE 7 Write the numbers below in descending order First compare the powers of 0. Both and. 0 are definitely larger than. 0 because 0 > 0. Since and. 0 have the same power of 0, we need to compare their decimal parts is larger because 9.8 >.. So in descending order, the numbers are 9.8 0,. 0,. 0. OR: Convert the numbers to decimal form to compare them.. 0 = = = Developmental Mathematics Book ISBN

17 exercise 7 08 Write in scientific notation. Select the correct answer A, B, C or D. A 9. 0 B 9. 0 C 9. 0 D 9 0 Write. 0 in decimal form. Select A, B, C or D. A B 0.00 C D 0.0 Copy and complete each statement. a = 0 b =. 0 c = 0 d = 0 e = 0 f = 0 Write each decimal in scientific notation. a b c d e 0.0 f g h Copy and complete each statement. a 8 0 = 0. b. 0 = 0. c. 0 0 = 0. d = 0. e 9. 0 = 0. f. 8 0 = 0. Write each number as a decimal. a. 0 b. 0 c 0 d.8 0 e 9 0 f 7. 0 g.08 0 h a Write the numbers below in descending order b List these same numbers in ascending order. 8 Write each number in scientific notation. a The thickness of a human hair is metres. b The length of an ant is m. c The amount of poison in a substance is grams. d The scale factor of a drawing is e The wavelength of an electron is m. 9 On the Internet, investigate the size of a micrometre and a nanometre. Shutterstock.com/Petruk Victor ISBN Chapter 7 Indices

18 7 09 Scientific notation on a calculator To enter a number in scientific notation on a calculator, use the or key. EXAMPLE 8 Use a calculator to write each number in decimal form. a b.70 0 a = On a calculator, enter:.8 9 = b.70 0 = On a calculator, enter:.70 ( ) = EXAMPLE 9 Evaluate each expression in scientific notation correct to two significant figures. a (. 0 ) ( ) b a On a calculator, enter:..8 8 = (. 0 ) ( ) = b On a calculator, enter: ( ) = = istockphoto/cemagraphics Developmental Mathematics Book ISBN

19 exercise 7 09 Use a calculator to write. 0 in decimal form. Select the correct answer A, B, C or D. A 000 B C 00 D Use a calculator to evaluate (.8 0 ) (7. 0 ) in decimal form. Select A, B, C or D. A B C D Use a calculator to write each number in decimal form. a.8 0 b. 0 c d e. 0 f g 0 7 h Evaluate each expression, correct to three significant figures where necessary. a.8 0 ( ) b 0 (. 0 8 ) c. 0 (. 0 ) d e (7 0 ) f. 0 (. 0 ) g 9. 0 ( 0 ) h 8 0 (.80 0 ) i.0 0 (7. 0 ) j Evaluate each expression, correct to two significant figures. a b c d e f g h Which number is larger? a. 0 or. 0 b. 0 or. 0 c 8 0 or 8 0 d 8. 0 or. 0 e 9. 0 or.7 0 f.8 0 or g. 0 or 7. 0 h 8. 0 or. 0 7 Evaluate each expression, correct to significant figures. a (. 0 ) b (.8 0 ) c. 0 d a If the mass of the planet Mercury is.0 0 kg and the mass of Venus is.87 0 kg, which is heavier and how many times heavier is it? b If the mass of Jupiter is kg and the mass of Saturn is.9 0 kg, which is heavier and how many times heavier is it? c If the mass of Earth is.97 0 kg and the mass of Neptune is.0 0 kg, which is heavier and how many times is it heavier? d If the mass of Mars is. 0 kg and the mass of Uranus is 8. 0 kg, which is heavier and how many times is it heavier? e From your answers in parts a to d above, order the planets from lightest to heaviest. ISBN Chapter 7 Indices

20 LANGUAGE ACTIVITY MIX AND MATCH Match each expression on the left with its simplified expression on the right. a a A a x x D x 8 (a ) E 8a x x 7 I a 8 (x ) M x a a N a 7 x x P x 9 8 (a ) T 9x 8 9 (x ) X x Match question numbers with answer letters to decode the answer to this riddle: What is always in a hurry and is found in a high position? Alamy/Cultura Creative (RF) Developmental Mathematics Book ISBN

21 PRACTICE TEST 7 Part A General topics Calculators are not allowed. Decrease $0 by %. If y =, evaluate 8 y. Find the average of 8 and. Copy this diagram and mark two vertically opposite angles. Part B Indices Calculators are allowed. 7 0 Multiplying terms with the same base Evaluate. 0 What percentage of $0 is $? 7 Convert 9 to -hour time. 8 Factorise 7p q pq. 9 Round 8.8 to two decimal places. 0 What is the probability of selecting a hearts card from a deck of playing cards? Simplify each expression. a a a 8 b 7 c x x d n n 7 0 Dividing terms with the same base Simplify each expression. a n n b 7 0 Power of a power 8x 8x 8 c 0x 8 x Simplify each expression using index notation. a ( ) b (m ) c (x ) d 7 0 Zero and negative indices n Simplify each expression. a 0 b x 0 c x 0 Write each expression with a positive index. a 7 b a c 7 0 Index laws review x Simplify each expression. a ab ab ( ) ( mn) b mn 8 c ( a ) a 9 0 ISBN Chapter 7 Indices

22 PRACTICE TEST Significant figures 7 Round each number to significant figures. a 70 b 08 c Scientific notation for large numbers 8 Write each number in scientific notation. a b c Scientific notation for small numbers 9 Write each number in scientific notation. a 0.08 b c Write these numbers in ascending order Scientific notation on a calculator Evaluate each expression. a b Developmental Mathematics Book ISBN