# Laws of Indices - Exeter Lesson 2- Laws of Indices Objectives To know what indices are To learn the rules of indices Oct 2011 INTO Foundation L2 What are Indices? Indices provide a way of writing numbers in a more compact and convenient form Indices is the plural of Index An Index is often referred to as a power Oct 2011 INTO Foundation L2 For example 5 x 5 x 5 = 53

2 x 2 x 2 x 2 = 24 7 x 7 x 7x 7 x 7 = 75 75 7 is the BASE NUMBER 75 & 24 are numbers in INDEX FORM Oct 2011 INTO Foundation L2 5 is the INDEX or POWER Combining numbers 5x5x5 x2x2x2x2 = 53 x 24 We can not write this any more simply Can ONLY combine BASE NUMBERS if they are the same Oct 2011

INTO Foundation L2 Rule 1 : Multiplication 26 x 2 4 = 210 24 x 2 2 = 26 35 x 3 7 = 312 General Rule Law 1 am x an = am+n Oct 2011

INTO Foundation L2 Rule 2 : Division 26 2 4 = 22 25 2 2 = 23 35 3 7 = 3-2 General Rule Law 2 am an = am-n Oct 2011 INTO Foundation L2

Rule 3 : Brackets (26)2 = 26 x 26 (35)3 = 3 5 x 3 5 x 35 = 212 = 315 General Rule Law 3 (am)n = am x n Oct 2011 INTO Foundation L2 Rule 4 : Index of 0 How could you get an answer of 30 ?

35 35 = 35-5 30 = 1 = 30 General Rule Law 4 a0 = 1 Oct 2011 INTO Foundation L2 Putting them together? 26 x 2 4 23 = 210 23

= 27 35 x 3 7 4 3 3 = 312 342 =2 = 38 25 x 2 24 x 22 Oct 2011 = 28 26 INTO Foundation L2

Works with algebra too! a6 x a 4 = a10 b5 x b7 = b12 c5 x c 3 c4 = c8 = c4 c4 -2 =a a5 x a3 a4 x a6 Oct 2011

= a8 a10 INTO Foundation L2 ..and a mixture 2a3 x 3a4 = 2 x 3 x a3 x a4 = 6a7 8a6 4a4 = (8 4) x (a6 a4) = 2a2 2 8a6 4a4 Oct 2011 2 INTO Foundation L2 = 2a2 Fractional indices (Using Law 1) We could write

1 1 2 x x x x But So Oct 2011 x x x INTO Foundation L2 1 2 x x 1 2

Fractional Indices Similarly 1 3 1 3 x x x x 3 3 1 3 3 x x x x 1 3

So x 3 x General Rule Law 5 1 n n a a Oct 2011 INTO Foundation L2 Negative Index Numbers. Simplify the expression below: 5 3 3

5 57 5 7 =5 Write the original expression again as a quotient: 555 5555555 Oct 2011 To understand this result fully consider the following:

-4 Expand the numerator and the denominator: Cancel out as many fives as possible: 1 5555 Write as a power of five: 1 4 5 Now compare the two results: INTO Foundation L2 Negative Indices The last Index rule

General Rule Law 6 a-m = 1 am Oct 2011 INTO Foundation L2 Summary Rule 1 : Multiplication of Indices. anxa m = Rule 2 : Division of Indices. a n a

m = . Rule 4 : For Powers Of Index Numbers. ( a m ) n = .. Rule 6 : For negative indices a -m =. Rule 5 : For fractional indices a1/n = na Oct 2011

INTO Foundation L2 Rule 3 : For Powers Of Index Numbers. a 0 = .. Exercises Section 2- Working with Indices Additional Questions if you get that far! Oct 2011 INTO Foundation L2 Travelling to Mars How long would it take a space ship travelling at an average speed of 2.6 103 km/h to reach Mars 8.32 107 km away? Oct 2011 INTO Foundation L2

Calculations involving standard form How long would it take a space ship travelling at an average speed of 2.6 103 km/h to reach Mars 8.32 107 km away? Rearrange speed = distance to give time time = 8.32 107 Time to reach Mars = 2.6 103

= 3.2 104 hours This is 8.32 2.6 Oct 2011 INTO Foundation L2 This is 107 103 distance speed Calculations involving standard form Use your calculator to work out how long 3.2 104 hours is in years. You can enter 3.2 104 into your calculator using the EXP key: 3 .

2 EXP 4 Divide by 24 to give the equivalent number of days. Divide by 365 to give the equivalent number of years. 3.2 104 hours is over 3 years. Oct 2011 INTO Foundation L2