Index Laws

What Is An Index Number. You should know that: 8 x 8 x 8 x 8 x 8 x 8 =8 6 We say“eight to the power of 6”. The power of 6 is an index number. The plural of index numbers is indices. What are the indices in the expressions below: (a) 3 x 5 4 (b) (c)  3 x  & 2 The number eight is the base number. If the index number is 1 we just write the base number Eg 8 1 =8

Multiplication Of Indices. We know that :7 x 7x 7 x 7 x 7 x 7 x 7 x 7 =7 8 But we can also simplify expressions such as : 6 3 x 6 4 To simplify: (1) Expand the expression.= (6 x 6 x 6) x (6 x 6 x 6 x 6) (2) How many 6’s do you now have? 7 (3) Now write the expression as a single power of 6. = 6 7 Key Result. 6 3 x 6 4 = 6 7

We can also simplify expressions such as :  2 x  3 To simplify: (1) Expand the expression. = (  x  ) x (  x  x  ) (2) How many  ’s do you now have? 5 (3) Now write the expression as a single power of . =  5 Key Result.  2 x  3 =  5 Multiplication Of Indices.

Using the previous example try to simplify the following expressions: (1) 3 7 x 3 4 = 3 11 (2)  5 x  9 =  14 (3) p 11 x p 7 x p 8 = p 26 We can now write down our first rule of index numbers: First Index Law: Multiplication of Indices. a n x a m = a n + m NB: This rule only applies to indices with a common base number. We cannot simplify  11 x p 7 as  and p are different bases.

What Goes In The Box ? 1 Simplify the expressions below : (1) 6 4 x 6 3 (2) g 7 x g 2 (3) d 6 x d (4) 14 9 x (5)  25 x  30 (6) 2 2 x 2 3 x 2 5 (7) p 7 x p 10 x p (8) 5 20 x 5 30 x 5 50 = 6 7 = g 9 = d 7 = =  55 = 2 10 = p 18 = 5 100

Division Of Indices. Consider the expression: The expression can be written as a quotient: Now expand the numerator and denominator. How many eights will cancel from the top and the bottom ? 4 Cancel and simplify. =8 3 Result: 8 7  8 4 = 8 3

Using the previous result simplify the expressions below: (1) 3 9  3 2 = 3 7 (2)  11   6 =  5 (3) p 24  p 13 = p 11 Second Index Law: Division of Indices. a n  a m = a n - m We can now write down our second rule of index numbers:

What Goes In The Box ? 2 Simplify the expressions below : (1) 5 9  5 2 (2) p 12  p 5 (3) 19 6  19 (4)  15   10 (5) b 40  b 20 (6) 2 32  2 27 (7) h 70  h 39 (8)  =5 7 = p 7 = 19 5 =  5 = b 20 = 2 5 = h 31 = 5 20

Zero Index Consider the expression: =1 Since the two results should be the same: 2 3  2 3 = 2 0 = 1 Using 2 nd Index Law

Zero Index Using the previous result simplify the expressions below: (1) 3 0 (2)  6   6 =  0 (3) (p 24 ) 0 = 1 Third Index Law : Zero Index a 0 = 1 ( where a  0 ) We can now write down our third rule of index numbers: = 1

Powers Of Indices. Consider the expression below: ( 2 3 ) 2 To appreciate this expression fully do the following: Expand the term inside the bracket. = ( 2 x 2 x 2 ) 2 Square the contents of the bracket. = ( 2 x 2 x 2 ) x (2 x 2 x 2 ) Now write the expression as a power of 2. = 2 6 Result: ( 2 3 ) 2 = 2 6

Use the result on the previous slide to simplify the following expressions: (1) ( 4 2 ) 4 (2) ( 7 5 ) 4 (3) ( 8 7 ) 6 = 4 8 = 7 20 = 8 42 We can now write down our fourth rule of index numbers: Fourth Index Law: For Powers Of Index Numbers. ( a m ) n = a m x n

Fifth and Sixth Index Laws These are really variations of the Fourth Index Law Fifth Index Law: (a x b) m = a m x b m Sixth Index Law:

Fifth and Sixth Index Laws - Variations Fifth Index Law: (2a x 3b) 2 = 2 2 a 2 x 3 2 b 2 = 4a 2 x 9b 2 = 36a 2 b 2 Sixth Index Law: Do not forget to raise the constants to the power as well? Eg:

Practice Maths Quest 10 Exercice 1A (page 5-6) Questions 1, 2, 3, 4 & 6: a, b, h, i Question 7: a to f

Negative Index Numbers. Simplify the expression below: 5 3  5 7 = 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:

The result on the previous slide allows us to see the following results: Turn the following powers into fractions: (1)(2)(3) We can now write down our seventh rule of index numbers: For Negative Indices:. a - m

More On Negative Indices. Simplify the expressions below leaving your answer as a positive index number each time: (1) (2)

What Goes In The Box ? 3 Change the expressions below to fractions: Simplify the expressions below leaving your answer with a positive index number at all times: (1)(2) (3) (4)(5)(6) (7)

What Goes In The Box ? 4 Simplify the expressions below leaving your answer as a positive index number. (1) (2)(3)(4) (5)(6)

Practice Maths Quest 10 : Exercise 1B (page 10-12) Questions 1, 2, 3 : 1 st column of each Question 4 : all Challenge questions Question