NUMBERS AND THE RULES OF ARITHMETIC
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1 MathScope. Mathematics for Egieerig ad Sciece Studets NUMBERS AND THE RULES OF ARITHMETIC. INDICES (POWERS OF NUMBERS). Notatio ( represets a umer) () is writte as ( raised to the power of or squared) () is writte as ( raised to the power of or cued) () is writte as ( raised to the power of or to the fourth) I geeral for a positive iteger... < terms > Here is said to e the power or idex of.. Multiplicatio () ( ) ( ) () ( ) ( ) I geeral 6 m m (A). Divisio () ( ) () 7 ( ) I geeral m m (B) Usig this result we fid for a value of that
2 MathScope. Mathematics for Egieerig ad Sciece Studets ut of course Therefore for a umer we have. Powers (C) ( ). 6 ( ).. 6 I geeral ( ) (D) m m. The Iterpretatio of Negative Powers Cosider the followig Which gives:. or (E) I other words, a umer raised to a egative power is equal to the iverse of the umer raised to the positive power..6 Fractioal Powers We defie m m m i.e. the th root of ad ( ). : (i) 8 so that 8 8 (ii) 8 so that 8 8 (iii) ( sice 6) ( ) or 6 6 (6) 8 (sice 8 6) As a result of the two defiitios aove it follows that the rules laelled (A) to (E) aove appl to ratioal umer idices as well as iteger idices. All these rules are collected here for ease of referece :
3 MathScope. Mathematics for Egieerig ad Sciece Studets I the followig is a real umer ad ad m ma e iteger or ratioal. (A) (E), m m (B) m (C) m (D) ( ) m m The followig examples illustrate what to do if a power operates o terms i rackets. (i) ( a) ( a) ( a) a a a From which we ca ifer results such as: ( ac) ( a) ( ) c ac 6 (ii) a a a a a From which we ca ifer results such as: ( a ) ( cd ) a a cd cd NB But otice the differece if a power operates o a additio (or sutractio). Terms like ( a ) or ( a ) or caot e writte i a simpler wa. True the first oe ma e chaged as follows ( a ) ( ) ( ) ( ) Distriutive Law a a a a a a a a ut this caot e descried as simplifig the expressio. Never, ever, ever make the mistake of writig ( a ) a.it is just plai wrog as ou will see if ou replace a ad umers. I the followig the oject is to simplif the give expressio as much as possile. (a). (). 7 (c). 8 (d) (e). 7 (f)
4 MathScope. Mathematics for Egieerig ad Sciece Studets (g) (h). ( ) (i). (j) ( ) (k) (l) x x x. Accordig to the rules we add the idices i the umerator ad sutract the idex i the deomiator which gives 6 x x x x.. This is a little more complicated so we proceed i two stages. First the comied idex i the umerator is. The comied idex i the deomiator is which must ow e sutracted from the umerator idex. This gives the aswer It is usual to express the fial result i positive powers so fiall If we summarise all this it gives a simple procedure for dealig with all prolems of this tpe ; The total comied idex of i the give expressio is ( ) as efore (m) ( ac ) ( ac ) ac 8 8. Whe as i this case there are umers ad several ac parameters ivolved simpl treat them all separatel. First the umers :- ( ) 8 8 6, ( 6), so the expressio ma e writte ac as. The comied idex of a is, of is 6 ac ad of c is
5 MathScope. Mathematics for Egieerig ad Sciece Studets ( ac ) So the result is ( ac ) 6 a a c 6 8 c () ( ) xz x z x z x z xz xz x z (o) a ( a ) a ( a ) Rememer that a term like ( a ) p q caot e simplified. However i this expressio we ca proceed as follows: a ( a ) a ( a ) ( a ) a ( a ) a ( a ) ( a ) Multiplig the umerator ad the deomiator of a expressio the same quatit leaves the value uchaged ad we ca ow comie the terms. a ( a ) ( a ) a ( a ) a ( a ) a a ( a ) ( a ) a ( a ) a ( a ) a ( a ) From which we fid the fial aswer to e a a Tutorial Evaluate the followig. Do ot use our calculator.. (a) () (c) 7 (d) 6. (a) ( ) () ( ) (c) (d). (a) () 7 8 (c) 7 (d) (. 6). (a) ( 7 ) () (c). (d) 7. 7
6 6 MathScope. Mathematics for Egieerig ad Sciece Studets Simplif the followig expressios. (a) 6. ) p. p p (c) x. x x ( t ) (d) (g) t. t ( a ) a( a ) ( a ) ( 6 8a c (e) ) 6 c (h) ( x z ) ( x z) (f) ( 8x ) ( 7x ) Click here to go to the solutios to this Tutorial. To retur to this spot after ou have looked at the solutios just use the ack utto. Click here to go ack to the list. 6
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