1. Lesson 2- Laws of Indices Objectives To know what indices are To learn the rules of indices Oct 2011INTO Foundation L2

2. 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 2011INTO Foundation L2

3. For example 5 x 5 x 5 = 5 3 2 x 2 x 2 x 2= 2 4 7 x 7 x 7x 7 x 7= 7 5 7 5 & 2 4 are numbers in INDEX FORM Oct 2011INTO Foundation L2 7575

4. Combining numbers 5 x 5 x 5 x 2 x 2 = 5 3 x 2 4 We can not write this any more simply Can ONLY combine BASE NUMBERS if they are the same Oct 2011INTO Foundation L2

5. Rule 1 : Multiplication 2 6 x 2 4 = 2 10 2 4 x 2 2 = 2 6 3 5 x 3 7 = 3 12 General Rule Law 1 a m x a n = a m+n Oct 2011INTO Foundation L2

6. Rule 2 : Division 2 6 ÷ 2 4 = 2 2 2 5 ÷ 2 2 = 2 3 3 5 ÷ 3 7 = 3 -2 General Rule Law 2 a m ÷ a n = a m-n Oct 2011INTO Foundation L2

7. Rule 3 : Brackets (2 6 ) 2 = 2 6 x 2 6 = 2 12 (3 5 ) 3 = 3 5 x 3 5 x 3 5 = 3 15 General Rule Law 3 (a m ) n = a m x n Oct 2011INTO Foundation L2

8. Rule 4 : Index of 0 How could you get an answer of 3 0 ? 3 5 ÷ 3 5 = 3 5-5 = 3 0 3 0 =1 General Rule Law 4 a 0 = 1 Oct 2011INTO Foundation L2

9. Putting them together? 2 5 x 2 3 2 4 x 2 2 = 2 8 2 6 2 6 x 2 4 2 3 = 2 10 2 3 3 5 x 3 7 3 4 = 3 12 3 4 = 2 7 = 3 8 = 2 2 Oct 2011INTO Foundation L2

10. Works with algebra too! a 5 x a 3 a 4 x a 6 = a 8 a 10 = a -2 a 6 x a 4 = a 10 b 5 x b 7 = b 12 c 5 x c 3 c 4 = c 8 c 4 = c 4 Oct 2011INTO Foundation L2

11. ..and a mixture… 2a 3 x 3a 4 = 6a 7 = 2 x 3 x a 3 x a 4 8a 6 ÷ 4a 4 = 2a 2 = (8 ÷ 4) x (a 6 ÷ a 4 ) 8a 6 4a 4 2 2 Oct 2011INTO Foundation L2 = 2a 2

12. Fractional indices (Using Law 1) We could write But So Oct 2011INTO Foundation L2

13. Fractional Indices Similarly General Rule Law 5 Oct 2011INTO Foundation L2

14. Negative Index Numbers. Simplify the expression below: 5 3  5 7 = 5 - 4 To understand this result fully consider the following: Write the original expression again as a quotient: Expand the numerator and the denominator: Cancel out as many fives as possible: Write as a power of five: Now compare the two results: Oct 2011INTO Foundation L2

15. Negative Indices The last Index rule a m General Rule Law 6 a -m = 1 Oct 2011INTO Foundation L2

16. Summary Rule 1 : Multiplication of Indices. a n x a 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 3 : For Powers Of Index Numbers. a 0 = ….. Rule 5 : For fractional indices a 1/n = n √a Oct 2011INTO Foundation L2

17. Exercises Section 2- Working with Indices Additional Questions if you get that far! Oct 2011INTO Foundation L2

18. Travelling to Mars How long would it take a space ship travelling at an average speed of 2.6 × 10 3 km/h to reach Mars 8.32 × 10 7 km away? Oct 2011INTO Foundation L2

19. Calculations involving standard form Time to reach Mars = 8.32 × 10 7 2.6 × 10 3 = 3.2 × 10 4 hours Rearrange speed = distance time time = distance speed to give This is 8.32 ÷ 2.6 This is 10 7 ÷ 10 3 How long would it take a space ship travelling at an average speed of 2.6 × 10 3 km/h to reach Mars 8.32 × 10 7 km away? Oct 2011INTO Foundation L2

20. Calculations involving standard form Use your calculator to work out how long 3.2 × 10 4 hours is in years. You can enter 3.2 × 10 4 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 × 10 4 hours is over 3½ years. Oct 2011INTO Foundation L2