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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 5 7 7 is the NUMBER 75 & 24 are numbers in INDEX FORM Oct 2011 INTO Foundation L2 5 is the 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 24 = 210 24 x 22 = 26 35 x 37 = 312 General Rule Law 1 am x an = am+n Oct 2011 INTO Foundation L2 Rule 2 : Division 26 ÷ 24 = 22 25 ÷ 22 = 23 35 ÷ 37 = 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 = 35 x 35 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 24 23 = 210 23 = 27 35 x 37 4 3 3 = 312 342 =2 = 38 25 x 2 = 28 24 x 22 26 Oct 2011 INTO Foundation L2 Works with algebra too! a6 x a4 = a10 b5 x b7 = b12 c5 x c 3 c4 = c8 = c4 c4 -2 a8 = a a5 x a3 = a4 x a6 a10 Oct 2011 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 28a6 4a4 Oct 2011 2 INTO Foundation L2 = 2a2 Fractional indices (Using Law 1) We could write 1 2 x x x x 1 But So Oct 2011 x xx INTO Foundation L2 1 2 x x 1 2 Fractional Indices Similarly 1 3 1 3 x x x x 1 3 x x x x 3 3 3 1 3 So x 3 x General Rule Law 5 1 n a n 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: 5 5 5 5 5 5 5 5 5 5 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 5 5 5 5 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 = n√a 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 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 3½ years. Oct 2011 INTO Foundation L2 4