1. Radians A Recap of last lesson’s work.

2. Indices Aims... Revise and Use Index Laws Understand the effect of negative indices. Use index laws to solve complicated indices problems

3. Learning Outcomes Name: To say what the index laws are. Describe: How to evaluate indices. How you use the index laws to simplify indices and expressions. How you can solve an equation with an unknown power using index laws. Explain: The effect of 0 or negative powers. How you can evaluate expressions that have indices in.

4. Independent Study Core 1 – Exercise 7B question 4 also 6 to 10. Next lesson we look at fractional indices see GCSE notes and/or mymaths lessons to get ahead. The Bakhshali Manuscript is an Ancient Indian (Modern day Pakistani) mathematical artefact. A leaf of birch bark found in a farmers field. Thought to date somewhere between 200-400 CE it contains the first evidence of the laws of indices.

5. Simplifying Indices You are going to be shown some problems that involve indices and how they can be simplified. We will see if you can spot the patterns involved. Are you ready? Lets go.

6. What Is… a 2 x a 4 a 6

7. What Is… b 5 x b 6 b 11

8. What Is… c 7 x c 8 c 15

9. What Is… y 15 x y 21 y 36

10. What Is… z 7 x z -5 z2z2

11. What Is… n -3 x n -4 n -7

12. What Is… a n x a m a n+m

13. What Is The First Index Law If you have two powers of the same base (number) that are multiplied this can be simplified by adding the powers. For indices with the same base... a n x a m = a n+m There are more; lets try something a little different. 1 minute

14. What Is… a 6 ÷ a 2 a4a4

15. What Is… b 12 ÷ b 3 b9b9

16. What Is… c 16 ÷ c 2 c 14

17. What Is… f 3 ÷ f 9 f -6

18. What Is… y -4 ÷ y -9 y5y5

19. What Is… a n ÷ a m a n-m

20. What Is The Second Index Law If you have two powers of the same base (number) that are divided this can be simplified by subtracting the powers. For indices with the same base... a n ÷ a m = a n-m There is one more which is an extension of the first… Like before one more time. 1 minute

21. What Is… Think a 5 x a 5 (a 5 ) 2 a 10

22. What Is… (b 8 ) 3 b 24

23. What Is… (c 2 ) 4 c8c8

24. What Is… (y 13 ) 13 y 169

25. What Is… (a n ) m a nm

26. What is the Third Index Law? Power of a power means the powers are multiplied. Using the first rule we can see this... (a n ) m a n x a n x a n... a n so that there is m lots of a n or a n+n+n… which is a nm To round up… Multiply... Add powers Divide... Subtract powers Power of powers... Multiply Powers 1 minute

27. Simplifying Indices & Algebra When we simplify multiplications; divisions and powers of terms that include indices we often need to apply the index laws. It is important that the index laws are applied correctly. The recall of orders of operations is also important.

28. Exercises Examples Simplify... a)3a 3 x (5a 4 ) 2 b) (3a 2 b 4 ) 3 ÷ (9a 3 b) Write the following in the form 2 n Time for an activity.

29. What About Other Powers? Zero/Negative Powers... Continue the dividing pattern (use fractions) a 0 = a -n = 2 -3 2 -2 2 -1 2020 21212 2323 248 3 -3 3 -2 3 -1 3030 3131 32323 3927 1 minute

30. What This Means Any powers on the bottom of a fraction can be moved into the numerator as negative powers (we usually leave the numbers as a single multiplier). E.g. It is important to use this later on in AS and A Level! 1 minute

31. Examples using negative indices Evaluate... ( 1 / 3 ) -3 x -3 if x = 2 Time for part 2.

32. Still One Topic To Do Lest you think these really easy here are some past paper questions… NOT DONE ANY LIKE THIS YET

33. How? Time to put it all together. 1 minute

34. Learning Outcomes Name: To say what the index laws are. Describe: How to evaluate indices. How you use the index laws to simplify indices and expressions. How you can solve an equation with an unknown power using index laws. Explain: The effect of 0 or negative powers. How you can evaluate expressions that have indices in.

35. Learning Outcomes Name: There are three index Laws a n xa m =a n+m ; a n ÷a m =a n+m ; (a n ) m =a nm Describe: Indices represent repeated multiplication. By applying the index laws to a problem with powers of the same unknowns you can simplify the expression. To solve equations with unknown powers you can write both sides as powers of one number then make the powers equal to each other and solve. Explain: Any value to the power of 0 is 1. Any negative power a -n is 1 over the number to the positive power 1 / a n. As with all algebra what works with numbers will work with expressions and indices are no different; you can substitute in powers or bases. E.g. x y if x=2 and y = 5 means x y =2 5 =32