Indices. Topic 1. Why learn this? What do you know? Learning sequence. number and algebra

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1 Topic Indices. Overview Why learn this? Don t you wish that your money could grow as quickly as a culture of acteria? Perhaps it can oth financial investments and a culture of acteria can grow exponentially, that is, according to the laws of indices. Indices are useful when a numer is continually multiplied y itself, ecoming very large, or perhaps very small. What do you know? Think List what you know aout indices. Use a thinking tool such as a concept map to show your list. pair Share what you know with a partner and then with a small group. share As a class, create a thinking tool such as a large concept map that shows your class s knowledge of indices. Learning sequence. Overview. Review of index laws. Negative indices. Fractional indices. Comining index laws.6 Review ONLINE ONLY

2 Watch this video The story of mathematics: Leiniz Searchlight ID: eles-80

3 . Review of index laws When a numer or pronumeral is repeatedly multiplied y itself, it can e written in a shorter form called index form. A numer written in index form has two parts, the ase and the index, and is written as: Base a x Index (power or exponent) Another name for an index is an exponent or a power. Performing operations on numers or pronumerals written in index form requires the application of the index laws. First Index Law: When terms with the same ase are multiplied, the indices are added. a m a n = a m + n Second Index Law: When terms with the same ase are divided, the indices are sutracted. a m a n = a m n WORKED EXAMPLE TI casio Simplify each of the following. a m n p m n p a a c x y 0x y THINK WRITE a Write the expression. a m n p m n p Multiply the terms with the same ase y adding the indices. Note: p = p. = m + n + p + = m 6 n 8 p Write the expression. a a Simplify y multiplying the coefficients, then multiply the terms with the same ase y adding the indices. = a + + = 6a 7 c Write the expression. c x y Simplify y dividing oth of the coefficients y the same factor, then divide terms with the same ase y sutracting the indices. 0x y = x y = x y Third Index Law: Any term (excluding 0) with an index of 0 is equal to. a 0 =, a 0 Maths Quest 0 + 0A

4 WORKED EXAMPLE Simplify each of the following. a ( ) 0 (a ) 0 THINK WRITE a Write the expression. a ( ) 0 Apply the Third Index Law, which states that any term (excluding 0) with an index of 0 is equal to. = Write the expression. (a ) 0 The entire term inside the rackets has an index of 0, so the racket is equal to. = Simplify. = Fourth Index Law: When a power (a m ) is raised to a power, the indices are multiplied. (a m ) n = a mn Fifth Index Law: When the ase is a product, raise every part of the product to the index outside the rackets. (a) m = a m m Sixth Index Law: When the ase is a fraction, multiply the indices of oth the numerator and denominator y the index outside the rackets. a a m = am m WORKED EXAMPLE TI casio Simplify each of the following. a (n ) (a 7 ) c a x y d ( ) THINK WRITE a Write the term. a (n ) Apply the Fourth Index Law and simplify. = n = n = 8n Write the expression. (a 7 ) Apply the Fifth Index Law and simplify. = a 7 = a 6 = 7a 6 Topic Indices

5 c Write the expression. c a x y Apply the Sixth Index Law and simplify. = x d Write the expression. d ( ) = 6x y 6 y Write in expanded form. = Simplify, taking careful note of the negative sign. = 6 Exercise. Review of index laws INDIVIdual PATHWAYS REFlecTIon Why are these laws called index laws? PracTISe Questions: a f, a f, a f, a f, 6, 7a f, 9, 0 consolidate Questions: d i, d i, a f, e l, 6, 7a f, 9, master Questions: d l, d l,, d o,, 6, 7d i, 8 6 Individual pathway interactivity int-6 doc-68 doc-69 FLUENCY WEa, Simplify each of the following. a a a a a a c d a a e m n 6 m n 7 f a c a c g mnp m n p h a a i a a j m mn 6m n k x xy 6x y l x y x x y WEc Simplify each of the following. a a a a 7 a c 6 d a7 a e 6 7 g m7 n m n h x y x y j 7a c a c k 0m n p 6m n p f 8m8 m i 6x 7 y 8x l x y z 8x y z WE Simplify each of the following. a a 0 () 0 c (m ) 0 d x 0 e 0 f (n) 0 g a 0 a a 0 h y 0 i x 0 (xy ) 0 6 Maths Quest 0 + 0A

6 WE Simplify each of the following. a (a ) (a ) c a m d a n e (a ) f (a ) g (m n ) j a m n h a m n i a a k a 7x y l a a m ( ) n ( 7) o ( ) MC a m 0 n is the simplified form of: A m n m n B 6m0 n D n(m ) n The value of (a) 0 is: n E a m n C (m n ) A B 9 C D E 6 MC a a a simplifies to: A 9a 8 B 0a 7 C 0a 8 D 9a 7 E a 8 x9 x 6 9x 0 x c simplifies to: A x 9 B 9x C x 9 D 9x 9 p 7 8q 9 simplifies to: p q A q D p q d 7a a 6 7 a simplifies to: a A 9a D a UNDERSTANDING 7 Evaluate each of the following. E x B E p q q B a 9 E a 9 C q C a a c ( ) d 6 e ( ) f Q R g 6 h ( ) 0 i ( ) 0 Topic Indices 7

7 <***dia***> <***dia***> numer and algera 8 Simplify each of the following. a (x y ) z a (p q ) 0 c m a n (mn) 0 e n m n p m q d a a x f (a m + n ) p reasoning 9 Explain why a a = a and not a 6. 0 Is x ever the same as x? Explain your reasoning using examples. Explain the difference etween x 0 and (x) 0. a In the following tale, enter the values of a and a when a = 0,, and. a 0 a a a + a a a Enter the values of a + a and a a in the tale. c What do you think will happen as a ecomes very large? Find algeraically the exact value of x if x+ = x. Justify your answer. Binary numers (ase numers) are used in computer operations. As the name implies, inary uses only two types of numers, 0 and, to express all numers. A inary numer such as 0 (read one, zero, one) means ( ) + (0 ) + ( 0 ) = = (in ase 0, the ase we are most familiar with). The numer 00 (read one, zero, one, zero) means ( ) + (0 ) + ( ) + (0 0 ) = = 0. If we read the inary numer from right to left, the index of increases y one each time, eginning with a power of zero. Using this information, write out the numers to 0 in inary (ase ) form. PROBLEM SOLVING Solve for x: a 7x 7 +x (7 x ) = x ( x ) = 8 Maths Quest 0 + 0A

8 ] numer and algera 6 For the following: a calculate the correct answer identify the error in the solution. a a c a a a c a = a c a a c = a c a ac = a ac 6 = a ac 6 = a 6 c 8 CHALLENGE.. Negative indices Consider the expression a This expression can e simplified in two different ways. a.. Written in expanded form: a a = a a a a a a a a = a a int-777 = a. Using the Second Index Law: a a = a = a So, a = a. Topic Indices 9

9 In general, a n = a0 a n ( = a0 ) = a 0 n (using the Second Index Law) = a n Seventh Index Law: a n = a n The convention is that an expression should e written using positive indices and with pronumerals given in alphaetical order. WORKED EXAMPLE Express each of the following with positive indices. a x m n c a THINK WRITE a Write the expression. a x Apply the Seventh Index Law. = x Write the expression. m n Apply the Seventh Index Law to write the expression with positive indices. c Write the expression and rewrite the fraction, using a division sign. Apply the Seventh Index Law to write the expression with positive indices. To divide the fraction, change fraction division into multiplication. = n m c = a a = a = a = a Part c from Worked example demonstrates the converse of the Seventh Index Law a n = an. WORKED EXAMPLE TI casio Simplify each of the following, expressing the answers with positive indices. a a a x y THINK xy c a m n WRITE a Write the expression. a a a Apply the First Index Law. Multiply terms with the same ase y adding the indices. = a + + = a Express the answer with positive indices. = a 0 Maths Quest 0 + 0A

10 Write the expression. x y Apply the Second Index Law. Divide terms with the same ase y sutracting the indices. xy Express the answer with positive indices. = x y c Write the expression. c a m n Apply the Sixth Index Law. Multiply the indices of oth the numerator and denominator y the index outside the rackets. Express all terms with positive indices. = = x y = x y = m 6 n m 6 n Simplify. = m 6 n Numers in index form can e easily evaluated if they are expressed with positive indices first. Consider the following example. WORKED EXAMPLE 6 Evaluate 6 without using a calculator. THINK WRITE Write the multiplication. 6 Apply the Seventh Index Law to write with a positive index. Multiply the numerator of the fraction y the whole numer. Evaluate the denominator. Cancel y dividing oth the numerator and denominator y the highest common factor (). = 6 = 6 = 6 7 = 9 Exercise. Negative indices INDIVIDUAL PATHWAYS Practise Questions: a i, a i, a f,, a e, 6a, 8a c, 9, a, consolidate Questions: a i, a i, c h,, a g, 6, 7, 8 e, 9, a,,,, 8 master Questions:, c o, c l,, d j, 6, 7, 8c f, 9 8 REFLECTION Are there any index laws from Section. that do not apply to negative indices? Individual pathway interactivity int-6 Topic Indices

11 FLUENCY WE Express each of the following with positive indices. a x y c a 9 d a e x y f m n g 6a c j 6a h a 6 k 7a i l a m n a WE Simplify each of the following, expressing the answers with positive indices. a a a x y x y c m n m n d a a 7 e xy 6 x y f x y 6xy g 6m n n m 6 h x y 9 x 7 y i m n 6m n j (a m ) k (p 7 q ) l (a ) m a p q n a a o a 6a WE6 Evaluate each of the following without using a calculator. a 6 c d e f 6 g j 6 h i k l 0 Write each of these numers as a power of. a 8 c d 8 6 Complete each statement y writing the correct index. a = 6 = c 7 = 7 d 6 = 6 e 0.0 = 0 f = 8 g 6 = h 6 = i 6 = j 6 = 8 6 Evaluate the following expressions. a a a c a d a 7 Write the following expressions with positive indices. a a a a a c a a 8 Evaluate each of the following, using a calculator. a 6 c 7 d a m n d a 8 e a 7 f (0.0) Maths Quest 0 + 0A

12 <***dia***> numer and algera 9 MC a x is the same as: c 8 A x B x C x D is the same as: a A a B a C a D is the same as: x E x a E a A B C D E 0 MC a Which of the following, when simplified, gives m n? A m n B m n C n D m E m n When simplified, a 7 a 6 is equal to: A a 6 B 9 a 6 C 9a c When (x 6 y ) is simplified, it is equal to: A x8 y B x 8 8y C y 8x 8 D a D 8y x 8 n m d If a ax y is equal to 89, then x and y (in that order) are: 6 a A and 6 B 6 and C and D and E and Understanding Simplify, expressing your answer with positive indices. a m n m n 6 E a E x 8 6y (m n ) 7 (m n ) c (a ) (a ) (a ) (a ) Simplify, expanding any expressions in rackets. a (r + s ) (r s ) (m + n ) c (xa+ ) x a+ x a(+) x Write a r 8 r r 6 in the form ar+. Write m m 6 m m m as a power of 6. Solve for x if x x = 8. d a px+ p x p 8(x+) (p x ) p (p x ) 0 Reasoning 6 Explain why each of these statements is false. Illustrate each answer y sustituting a value for the pronumeral. a x 0 = 9x x = x c a a 7 = a d c = c Topic Indices

13 PROBLEM SOLVING 7 Solve for x and y if x y = 6 and x y =. Hence, evaluate x 7 y y. 8 Solve for n. Verify your answers. a ( n ) n ( n ) =. Fractional indices (n ) n ( n ) 8 Terms with fractional indices can e written as surds, using the following laws:. a n = " n a m. a n = " n a m = " n a m To understand how these laws are formed, consider the following numerical examples. We know = and that!! =!6 = It follows, then, that =!. Similarly, we know that = 8 and that " 8 " 8 " 8 = " = 8 It follows, then that 8 = " 8. This oservation can e generalised to a n = " n a. m Now consider: a n = a m n = (a m n ) = " n a m or n n am = a m = a n m = (" n a) m Eighth Index Law: am n = " n a m = (" n a) m = WORKED EXAMPLE 7 Evaluate each of the following without using a calculator. a 9 6 THINK WRITE a Rewrite the numer using the Eighth Index Law. a 9 =!9 Evaluate. = Rewrite the numer using n am = (" n a) m. 6 = (!6) = Simplify and evaluate the result. = 6 Maths Quest 0 + 0A

14 WORKED EXAMPLE 8 a m m TI casio Simplify each of the following. (a ) 6 c ± x THINK y WRITE a Write the expression. a m m Apply the First Index Law to multiply terms with the same ase y adding the indices. = m Write the expression. (a ) 6 Use the Fourth Index Law to multiply each index inside the rackets y the index outside the rackets. = a Simplify. = a c Write the expression. c ± x y 6 6 Use the Sixth Index Law to multiply the index in oth the numerator and denominator y the index outside the rackets. = x y8 Exercise. Fractional indices INDIVIDUAL PATHWAYS Practise Questions:, 6a, d, g, 7a, d, 8a, d, g, 9a, d, 0a, d, g, a, d, g,,, a, d, g,, 6 consolidate Questions:, 6a,, e, h, i, 7a,, c, f, 8a,, d, e, g, h, 9a,, d, e, 0, e, h,, e, h,,,, e, h,, 6, 7 master Questions:, 6c, f, i, 7c, f, 8c, f, i, 9, c, e, f, 0c, f, i, c, f, i, 9 REFLECTION Why is it easier to perform operations with fractional indices than with expressions using surds? Individual pathway interactivity int-6 FLUENCY WE7 Evaluate each of the following without using a calculator. a 6 c 8 d 8 e 6 f 8 Write the following in surd form. a m c 7 d 7 e w8 f w. g h a 0. Write the following in index form. a!t " 7 c " 6 6 d " 7 x 6 0 x e " 6 x 7 f " w 0 g "w h " n doc-76 doc-77 doc-78 doc-79 Topic Indices

15 Without using a calculator, find the exact value of each of the following. a 8 8 c d e f 7 g 7 h 8 6 i 0 j 6 k 7 l Using a calculator, evaluate each of the following. Give the answer correct to decimal places. a c 7 d 89 e 8 f (0.6) g a h a i a 6 WE8a Simplify each of the following. a 8 8 c a a d x x e m m f 7 7 g y y9 h a8 0.0a i x x 7 Simplify each of the following. a a a x y 9 x y c a a d 6m 7 m n e x y z x 6 y z f a 8 c c 8 Simplify each of the following. a c 6 d a 7 a7 e x x f m m9 g x 7n h i x n 0 9 Simplify each of the following. a x y x d 0x y x y a 9 y a e 0a 0 Simplify each of the following. a d (a ) 0 e m g p 7 a 8 9 h xm n n p c m 8 n 7 p 8 q f 7p q6 c 7 6 f i 7 n a c m 8 6 Maths Quest 0 + 0A

16 WE8, c Simplify each of the following. a a d a g ± m c 7 n8 MC a y is equal to: A k c A y B y (a ) c x e x h ± y z c9 is not equal to: k B " k C k " g is equal to: A g B g C g 7 y 8 f a a i ± x7 y C (y ) D " y E y m MC a If a n is equal to a, then m and n could not e: D " k E (k ) D g E g A and B and 6 C and 8 D and 9 E oth C and D m When simplified, q a n r m a p n m n p p p m a A B n m Simplify each of the following. n is equal to: C mp n a n m D a p m E m np a nm a "a 8 " 9 c " m 6 d "6x e " 8y 9 f " 6x 8 y g " 7m 9 n h " p q 0 i " 6a 6 8 UNDERSTANDING The relationship etween the length of a pendulum (L) in a grandfather clock and the time it takes to complete one swing (T) in seconds is given y the following rule. Note that g is the acceleration due to gravity and will e taken as 9.8. T = πa L g a Calculate the time it takes a m long pendulum to complete one swing. Calculate the time it takes the pendulum to complete 0 swings. c How many swings will e completed after 0 seconds? REASONING 6 Using the index laws, show that " a 0 = a. p Topic Indices 7

17 7 To rationalise a fraction means to remove all non-rational numers from the denominator a of the fraction. Rationalise y multiplying the numerator and denominator y + "!, and then evaluate if = a and a =. Show all of your working. PROBLEM SOLVING doc-80 8 Simplify m m n + n p. m n p 9 A scientist has discovered a piece of paper with a complex formula written on it. She thinks that someone has tried to disguise a simpler formula. The formula is: " a a " a "a "a a a a! a Simplify the formula using index laws so that it can e worked with. From your simplified formula, can a take a negative value? Explain. c What is the smallest value for a for which the expression will give a rational answer? Consider only integers.. Comining index laws When several steps are needed to simplify an expression, expand rackets first. When fractions are involved, it is usually easier to carry out all multiplications first, leaving one division as the final process. Final answers are conventionally written using positive indices. WORKED EXAMPLE 9 Simplify each of the following. a (a) 6a n 9 n+ 8 n THINK a Write the expression. a Apply the Fourth Index Law to remove the racket. Apply the Second Index Law for each numer and pronumeral to simplify. WRITE (a) 6a = 6a 6a = 8a Write the answer. = 8a Write the expression. n 9 n+ Rewrite each term in the expression so that it has a ase of. Apply the Fourth Index Law to expand the rackets. 8 n = n ( ) n+ ( ) n = n n+ n 8 Maths Quest 0 + 0A

18 Apply the First and Second Index Laws to simplify and write your answer. = n n = n WORKED EXAMPLE 0 Simplify each of the following. a (a ) a THINK 7xy (x y ) c m n m 7 n 7m n mn WRITE a Write the expression. a (a ) a Apply the Fourth Index Law. Multiply each index inside the rackets y the index outside the rackets. = a a Evaluate the numer. = 6a a Multiply coefficients and multiply pronumerals. Apply the First Index Law to multiply terms with the same ase y adding the indices. Write the expression. Apply the Fourth Index Law in the denominator. Multiply each index inside the rackets y the index outside the rackets. Apply the Second Index Law. Divide terms with the same ase y sutracting the indices. Use a m = am to express the answer with positive indices. = 6 a + + = 6a 7 7xy (x y ) = 7xy 9x 6 y = 7x y 9 = 7 9x y c Write the expression. c m n m 7 n Simplify each numerator and denominator y multiplying coefficients and then terms with the same ase. Apply the Second Index Law. Divide terms with the same ase y sutracting the indices. 7m n mn = 6m n 7m n = 6m8 n 0 7 Simplify the numerator using a 0 =. = 6m8 7 = 6m8 7 Topic Indices 9

19 WORKED EXAMPLE TI Simplify each of the following. casio a (a ) a 0 a (a ) 7 8m n (6mn ) THINK m n 6m n WRITE a Write the expression. a (a ) a 0 a Remove the rackets in the numerator of the first fraction and in the denominator of the second fraction. Multiply the numerators and then multiply the denominators of the fractions. (Simplify across.) Divide terms with the same ase y sutracting the indices. (Simplify down.) (a ) 7 = a 6 a 0 a a 7 = a6 a 7 = a Express the answer with positive indices. = a Write the expression. 8m n (6mn ) m n 6m n Remove the rackets. = 8m n 6m n m n 6 6m n Change the division to multiplication. = 8m n 6m n 6m n 6 m n Multiply the numerators and then multiply the denominators. (Simplify across.) Cancel common factors and divide pronumerals with the same ase. (Simplify down.) 6 Simplify and express the answer with positive indices. = 8m n 86mn = m n 8 = 8m n Note that the whole numers in part of Worked example could e cancelled in step. Exercise. Comining index laws INDIVIDUAL PATHWAYS REFLECTION Do index laws need to e performed in a certain order? Practise Questions: a d, a d, a d, a d, a d, 6, 7, 9, 0, a, d, consolidate Questions: c h, c f, c g, f, c f, 6 0, e, master Questions: f j, e i, f i, d f, e h, 6 0, c f, Individual pathway interactivity int-6 0 Maths Quest 0 + 0A

20 FLUENCY WE0a Simplify each of the following. a (a ) a (a ) a 6 c m n (m n ) 6 d (pq ) (p q ) e (a 7 ) (a ) g 6x i y p x q y f ( c ) (c ) h 6m n m p q j 8p q 6p n q WE0 Simplify each of the following. a a (a ) x y 6 (xy ) c (m n ) (m n ) 7 d a x y 0 x 7 y 6 e a (a 7 ) f a g h g h g i p 6 q p x q y z x y z h a c c WE0c Simplify each of the following. a a a a m6 n mn 6m 7 n 6 c 0m6 n m n m n m n d 6x y x 6 y 9xy x y 6 e (6x y ) 9x y xy 7 f x y xy 0x y x y g a (a ) 6(a ) a h (p 6 q ) pq p q (pq ) i 6x y x y x y x y Topic Indices

21 WEa Simplify each of the following. a a a a6 7 a 9 (a6 ) 0a 7 a6 6a c (m n ) (m 6 n) (m n ) (mn) d a m n mn 6m n m n 0 e a xy x y a x y 9 y 0 f x y (x y ) x y 6 x 7 y g p6 q q a p6 q p h a 6a a a i x 9x y y x x y WE Simplify each of the following. a a 6a 7 a9 a 6 7a a 6 7 a a a 6 c a a9 6 a a7 d x y 6 x y x6 y 0xy e a x y xy x 6 y 0 (x y ) f m n m 6 n am n 6 m n g m n 6m 8m n n h q c 6c r c UNDERSTandIng 6 Evaluate each of the following. a ( ) 0 ( 0 ) ( 6 ) ( ) (6 9 ) 0 6 ( ) 7 Evaluate the following for x = 8. (Hint: Simplify first.) (x) a x x ( ) a 8 a Simplify the following fraction. y 9 y (a) y (a y ) ( y ) Find the value of y if the fraction is equal to. Maths Quest 0 + 0A

22 9 MC Which of the following is not the same as (xy)? A 8x y B (!xy) C "6x y E xy (xy ) D (x y ) (!) 0 MC The expression x y (xy ) xy is equal to: 0 6x A x y 6 B x 6 C x y 6 D xy 6 E 8xy Simplify the following. a " m n "mn g h a n c 9 d 6 e a a a a a f " d " d REASONIng In a controlled reeding program at the Melourne Zoo, the population (P) of koalas at t years is modelled y P = P 0 0 kt. The initial numer of koalas is 0 and the population of koalas after year is 0. Given P 0 = 0 and k = 0.: a calculate the numer of koalas after years determine when the population will e equal to 000. The decay of uranium is modelled y D = D 0 kt. If it takes 6 years for the mass of uranium to halve, find the percentage remaining after: a years years c 0 years. Give your answers to the nearest whole numer. Topic Indices

23 doc-8 PROBLEM SOLVING Simplify 7x+ 7 x x. Simplify z + z. z + z CHALLENGE. Maths Quest 0 + 0A

24 ONLINE ONLY.6 Review The Maths Quest Review is availale in a customisale format for students to demonstrate their knowledge of this topic. The Review contains: Fluency questions allowing students to demonstrate the skills they have developed to efficiently answer questions using the most appropriate methods Prolem Solving questions allowing students to demonstrate their aility to make smart choices, to model and investigate prolems, and to communicate solutions effectively. A summary of the key points covered and a concept map summary of this topic are availale as digital documents. Review questions Download the Review questions document from the links found in your ebookplus. Language int-86 int-87 ase constant denominator evaluate exponent expression index index law negative numerator positive power indices pronumeral simplify sustitute surd int-88 Link to assesson for questions to test your readiness FOR learning, your progress AS you learn and your levels OF achievement. assesson provides sets of questions for every topic in your course, as well as giving instant feedack and worked solutions to help improve your mathematical skills. The story of mathematics is an exclusive Jacaranda video series that explores the history of mathematics and how it helped shape the world we live in today. Leiniz (eles-80) tells the story of Gottfried Leiniz, a remarkale mathematician who helped refine the inary system that underpins nearly every piece of modern technology in the world today. Topic Indices

25 numer <INVEStigation> investigation and algera for rich task or <numer and algera> for puzzle RICH TASK Digital world: A it of this and a yte of that Complete the tale elow to show the difference in value etween the inary and decimal systems. 6 Maths Quest 0 + 0A

26 The two numering systems have led to some confusion, with some manufacturers of digital products thinking of a kiloyte as 000 ytes rather than 0 ytes. Similar confusion arises with megaytes, gigaytes, teraytes and so on. This means you might not e getting exactly the amount of storage that you think. If you ought a device quoted as having 6 GB memory, what would e the difference in memory storage if the device had een manufactured using the decimal value of GB as opposed to the inary system? Many devices allow you to check the availaility of storage. On one such device, the iphone, availale storage is found y going to General under the heading Settings. How much storage is left in MB on the following iphone? If each photo uses. MB of memory, how many photos can e added? General Usage Storage.9 GB Availale 9. GB Used Photos & Camera.6 GB Radio.6 GB Maps. GB My Movie 6 MB Have you ever wondered aout the capacity of our rain to store information and the speed at which information is transmitted inside it? Discuss how the storage and speed of our rains compares to our current aility to send and store information in the digital world. The capacity of the human rain is 0 00 teraytes. On average 0 million illion its of information are transmitted within the rain per second. 6 Investigate which country has the fastest internet speed and compare this to Australia. Topic Indices 7

27 <INVEStigation> numer and algera for rich task or <numer and algera> for puzzle CODE PUZZLE What historical event took place in France in 78? Match the expressions across the top and on the left-hand side with the equivalent answer along the ottom or right-hand side y ruling a line etween the dots. Each line passes through a numer and a letter to reveal the puzzle's code. x y 9x y 6x y 6x y x (x y) 0 x (x ) (x ) ( ) xy x y 8 x (xy ) 0 X x x x 6x y 6 Y 6 7 A S H V T N 6 M L 0 9 I G 8 B O F R 7 D x y E x x 7 xy 8x 9 xy 6x y x 0 x y x 8 x 6y x x 6 x Maths Quest 0 + 0A

28 Activities. Overview Video The story of mathematics (eles-80). Review of index laws Digital docs SkillSHEET (doc-68): Index form SkillSHEET (doc-69): Using a calculator to evaluate numers given in index form Interactivity IP interactivity. (int-6): Review of index laws. Negative indices Interactivities Colour code reaker (int-777) IP interactivity. (int-6): Negative indices. Fractional indices Digital docs SkillSHEET (doc-76): Addition of fractions SkillSHEET (doc-77): Sutraction of fractions To access ebookplus activities, log on to SkillSHEET (doc-78): Multiplication of fractions SkillSHEET (doc-79): Writing roots as fractional indices WorkSHEET. (doc-80): Fractional indices Interactivity IP interactivity. (int-6): Fractional indices. Comining index laws Digital doc WorkSHEET. (doc-8): Comining index laws Interactivity IP interactivity. (int-6): Comining index laws.6 Review Interactivities Word search (int-86) Crossword (int-87) Sudoku (int-88) Digital docs Topic summary (doc 80) Concept map (doc 80) Topic Indices 9

29 Answers Topic Indices Exercise. Review of index laws a a 7 a 6 c 8 d a 7 e m n f a 7 c g m 6 n p h 6a i 0a 9 j 6m 8 n 7 k x 6 y 6 l x 8 y 6 a a a c d a e f m g m n h y i x y j 7 k m p l a c d e f g h 7 i a a 6 6a 0 c 8 m8 d 9 n8 e a 6 f 9a 6 g 6m n 0 h 7 6 m6 n i j 6m n 8 k x 8y l a 6 8a 6 xy m n 9 o a D D 6 a C E c B d D 7 a 6 7 c 6 d 8 e 600 f 7 g 0 h i 8 a x yz a c m a n d a x e n p m q f a mp + np x 9 a = a a a a = a a a a = a a a a a = a, not a 6 Explanations will vary. 0 They are equal when x =. Explanations will vary. x 0 = and (x) 0 =. Explanations will vary. a, a 0 a 0 7 a 0 0 a + a 0 8 a a c a a will ecome much larger than a + a. ±! a x = x = 0, 6 a a c 7 The student made a mistake when multiplying the two rackets in line. Individual rackets should e expanded first. Challenge..08 seconds Exercise. Negative indices a d x a y e x y g 6a h a 6 i c j a k 7 a a 6 a x 6 y e y y f x 6x i j m n a m 0 a m 7q9 8p 6 e i = n f 6 a c g n 8 m n k q8 p o 8a 6 6 c 8 c f l a 9 m n a m a n d a h y x l a 8 d 8 9 g 8 h 7 j k l a c d 6 a c d e f 0 g h i 6 j 6 a 7 a 8 a a 79 c or e a 6807 c 7 c a 0 76 d 6 f d 9 a D C c B 0 a B D c C d E a m n 8 n m c a 7 6 a r 6 s 6 m 0 + m n + n 0 c d p r 6 m x = 6 Answers will vary; check with your teacher. 7 x =, y = ; 7 8 a n =, n =, d m n Exercise. Fractional indices a c 9 d e f a! " m c " 7 d "7 e " 8 w f " w g " 0 0 h "a 0 Maths Quest 0 + 0A

30 a t 7 c d x7 7 e x6 f w g w n h x a 6 c 8 d 6 e f 9 g 9 h 7 i 000 j 6 k!7 l " a.. c.8 d.6 e. f 0.66 g 0. h 0.8 i a c a6 d x0 8 e 0m f 7 0 g y 9 9 h 0.0a8 i 7 x 7 a a x 9 d m n e x y 9 6 y 6 z 9 c 6a 6 f 8a 8 a 6 c d a7 e x g x 0 h n i c f m a x y a c m8 n6 d x y e 7 a0 0 f p q 7 0 a c 7 d a0 e m6 f 6 g p p h xm i c c ma 6 7 a a 6 a c x y d a 9 c e x y z f 8 g m 7 n h 8 c 7 a E C c B a E B i a 7 x a a c m d x e y f x y g m n h pq i 6a 6 a.007 s 0.07 s c.98 swings a Q " 6 ( a 0 ) = a R 7 ; 9 8 m n + p 9 a a No, ecause you can t take the fourth root of a negative numer. c a = y 8 Exercise. Comining index laws a a 0 9 8a 6 c n d 00p 8 q 8 m 9 e 6a 0 0 i a 6 7 p 8a 7 f j c 6 7 8p q8 x y 6 7 g x 8 y d 6y6 e a 7 f x g p q h a a d x y 8 g a a 7 a h 8m c 7 8m 9 n 6 7h 8g c 8 i x y z 8n c m n e 6x6 y a a 7 d m 9n g p q 9 e 6 a h 7q i x p e f c y x y0 n 9 m 9 f 8x y 6 8x y h a 6a 6 a 8 y 6 7 c 0 8a f 6m 9 n 9 g 6m 7x a y y = 9 E 0 A a m 6 n 76 6 m or Å n 7 d or n i x y0 d h n 8x y 7 c0 g 6 h n c e a 6 8 or a6 f d or "d 8 a 79 koalas During the 6th year a 79% 6% c % z + z + " Challenge. x = y y z Topic Indices

31 Investigation Rich task Unit Symol Power of and value in ytes Byte B 0 = 0 0 = Kiloyte KB 0 = 0 0 = 000 Power of 0 and value in ytes Megayte MB 0 = = Approximately. GB 99.6 MB 8 photos Discuss with your teacher. 6 Discuss with your teacher. The discussion will depend on the latest information from the internet. Code puzzle The first manned hot-air alloon flight in a alloon invented y the Montgolfier rothers Gigayte GB 0 = = Terayte TB 0 = = Maths Quest 0 + 0A

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