Indices or Powers - mathcentre.ac.uk

1 Indices or Powersmc-TY-indicespowers-2009-1A knowledge of Powers , or Indices as they are often called, isessential for an understandingof most algebraic processes. In this section of text you willlearn about Powers and rules formanipulating them through a number of worked order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: simplify expressions involving Indices use the rules of Indices to simplify expressions involving Indices use negative and fractional first rule:am an=am+ second rule:(am)n= third rule:am an=am fourth rule:a0= fifth rule:a 1=1aanda m= sixth rule:a12= aanda1q=q final result:apq= (ap)1q=q ap,apq= (a1q)p= (q a) mathcentre 20091.

2 IntroductionIn the section we will be looking atindicesorpowers. Either name can be used, and both namesmean the same , they are a shorthand way of writing multiplications of , suppose we have4 4 4We write this as 4 to the power 3 :43So4 4 4 = 43 The number 3 is called thepowerorindex. Note that the plural of index is PointAn index, or power , is used to show that a quantity is repeatedly multiplied by can be done with letters as well as numbers. So, we might have:a a a a aSince there are fivea s multiplied together we write this as ato the power 5 .a5 Soa a a a a= if we had2x2raised to the power 4 ? This means four factors of2x2multiplied together,that is,2x2 2x2 2x2 2x2 This can be written2 2 2 2 x2 x2 x2 x2which we will see shortly can be written of a power or index is simply a form of notation, that is, a way of writing something mathematicians have a way of writing things down they like to use their notation in otherways.

3 For example, what might we mean bya 2ora12ora0?To proceed further we needrulesto operate with so we can find out what these notationsactually mathcentre 2009 Exercises1. Evaluate each of the )35b)73c)29d)53e)44f)832. The first ruleSuppose we havea3and we want to multiply it bya2. That isa3 a2=a a a a aAltogether there are fivea s multiplied together. Clearly, this is the same asa5. This suggestsour first first rule tells us that if we are multiplying expressionssuch as these then we add the indicestogether. So, if we haveam anwe add the Indices to getam an=am+nKey Pointam an=am+n3. The second ruleSuppose we hada4and we want to raise it all to the power 3. That is(a4)3 This meansa4 a4 a4 Now our first rule tells us that we should add the Indices together.

4 So that isa12 But note also that 12 is4 3. This suggests that if we haveamall raised to the powerntheresult is obtained by multiplying the two Powers to getam n, or mathcentre 2009 Key Point(am)n=amn4. The third ruleConsider a3=a7a3=a a a a a a aa a aWe can now begin dividing out the common factors ofa. Three of thea s at the top and thethreea s at the bottom can be divided out, so we are now left witha41that isa4 The same answer is obtained by subtracting the Indices , thatis,7 3 = 4. This suggests ourthird rule, thatam an=am Pointam an=am n5. What can we do with these rules ? The fourth ruleLet s have a look ata3divided bya3. We know the answer to this. We are dividing a quantityby itself, so the answer has got to be a3= 1 Let s do this using our rules; rule 3 will help us do this.

5 Rule3 tells us that to divide the twoquantities we subtract the Indices :a3 a3=a3 3=a0We appear to have obtained a different answer. We have done thesame calculation in twodifferent ways. We have done it correctly in two different ways. So the answers we get, even ifthey look different, must be the same. So, what we have isa0= mathcentre 2009 Key Pointa0= 1 This means that any number raised to the power zero is 1. So20= 1(1,000,000)0= 1(12)0= 1( 6)0= 1 However, note that zero itself is an exception to this be evaluated. Any number,apart from zero, when raised to the power zero is equal to The fifth ruleLet s have a look now at doing a division a7=a3a7=a a aa a a a a a aAgain, we can now begin dividing out the common factors ofa.

6 The 3a s at the top and threeof thea s at the bottom can be divided out, so we are now left witha3 a7=1a a a a=1a4 Now let s use our third rule and do the same calculation by subtracting the a7=a3 7=a 4We have done the same calculation in two different ways. We have done it correctly in twodifferent ways. So the answers we get, even if they look different, must be the same. So1a4=a 4So a negative sign in the index can be thought of as meaning 1 over .Key Pointa 1=1aand more generallya m= mathcentre 2009 Now let s develop this further in the following the next two examples we start with an expression which hasa negative index, and rewrite itso that it has a positive index, using the rulea m= 2=122=145 1=151=15We can reverse the process in order to rewrite quantities so that they have a negative 1172= 7 2 One you should try to remember is1a=a 1as you will probably use it the now what about an example like17 2.

7 Using the Example above, we see that this means11/72. Here we are dividing by a fraction, and to divide by a fraction we need to invert andmultiply so:17 2=11/72= 1 172= 1 721= 72 This illustrates another way of writing the previous keypoint:Key Point1a m=amExercises2. Evaluate each of the following leaving your answer as a proper )2 9b)3 5c)4 4d)5 3e)7 3f)8 37. The sixth ruleSo far we have dealt with integer Powers both positive and negative. What would we do if we hada fraction for a power , likea12. To see how to deal with fractional Powers consider the mathcentre 2009 Suppose we have two identical numbers multiplying togetherto give another number, as in, forexample7 7 = 49 Then we know that 7 is a square root of 49.

8 That is, if72= 49then7 = 49 Now suppose we found thatap ap=aThat is, when we multipliedapby itself we got the resulta. This means thatapmust be a squareroot , look at this another way: noting thata=a1, and also that, from the first rule,ap ap=a2pwe see that ifap ap=athena2p=a1from which2p= 1and sop=12 This shows thata1/2must be the square root ofa. That isa12= aKey Pointthe power 1/2 denotes a square root:a12= aSimilarlya13=3 athis is the cube root ofaanda14=4 athis is the fourth root ofaMore generally,Key Pointa1q=q mathcentre 2009 Work through the following examples:ExampleWhat do we mean by161/4?For this we need to know what number when multiplied togetherfour times gives 16. The answeris 2.

9 So161/4= do we mean by811/2? For this we need to know what number when multiplied by itselfgives 81. The answer is 9. So8112= 81 = about24315? What number when multiplied together five times gives us 243? If we arefamiliar with times-tables we might spot that243 = 3 81, and also that81 = 9 9. So2431/5= (3 81)1/5= (3 9 9)1/5= (3 3 3 3 3)1/5So3multiplied by itself five times equals 243. Hence2431/5= 3 Notice in doing this how important it is to be able to recognise what factors numbers are madeup of. For example, it is important to be able to recognise that:16 = 24,16 = 42,81 = 92,81 = 34and so will find calculations much easier if you can recognise innumbers their composition as powersof simple numbers such as 2, 3, 4 and 5.

10 Once you have got these firmly fixed in your mind, thissort of calculation becomes Evaluate each of the )1251/3b)2431/5c)2561/4d)5121/9e)3431/3f )5121/38. A final resultWhat happens if we takea34?We can write this as follows:a34= (a14)3using the 2nd rule(am)n=amnExampleWhat do we mean by1634?1634= (1614)3= (2)3= mathcentre 2009We can also think of this calculation performed in a slightlydifferent way. Note that instead ofwriting(am)n=amnwe could write(an)m=amnbecausemnis the same do we mean by823? One way of calculating this is to write823= (813)2= (2)2= 4 Alternatively,823= (82)13= (64)13= 4 Additional noteDoing this calculation the first way is usually easier as it requires recognising Powers of smaller example, it is straightforward to evaluate275/3as275/3= (271/3)5= 35= 243because, at least with practice, you will know that the cube root of 27is 3.