This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 0 6 ÷ 10 2 (f) 5 4 ÷ 52 (a) Without using a calculator, write down the values of k and m. 64 = 8 2 = 4 k = 2 m
(b) Complete the following: 2 15 = 32 768
2 14 =
(KS3/99Ma/Tier 57/P1) 3.2 Laws of Indices
There are three rules that should be used when working with indices:
When m and n are positive integers,
1. am × an = am + n 2. a ÷a = a 3. (a ) m m n n m−n am
or n = a m − n
a ( m ≥ n) = am × n These three results are logical consequences of the definition of a n , but really
need a formal proof. You can 'verify' them with particular examples as below, but
this is not a proof:
2 7 × 2 3 = (2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (2 × 2 × 2) = 2×2×2×2×2×2×2×2×2×2
= 2 10 (here m = 7, n = 3 and m + n = 10 )
40 MEP Y9 Practice Book A or,
27 ÷ 23 = 2×2×2×2×2×2×2
2×2×2 = 2×2×2×2
= 24 (2 )
7 Also, 3 (again m = 7, n = 3 and m − n = 4 ) = 27× 27× 27
= 2 21 (using rule 1) (again m = 7, n = 3 and m × n = 21) The proof of the first rule is given below: Proof
am× an = a × a × ... × a × a × a × ... × a
14 244
4
3
14 244
4
3
m of these
n of these = a × a × ... × a × a × a ... × a
14444 244444
4
3
(m + n) of these = am+n The second and third rules can be shown to be true for all positive integers m and
n in a similar way.
We can see an important result using rule 2:
xn
= xn − n = x0
n
x
but xn
= 1, so
xn x0 = 1
This is true for any nonzero value of x, so, for example, 30 = 1, 270 = 1 and 10010 = 1 . 41 MEP Y9 Practice Book A 3.2
Example 1 Fill in the missing numbers in each of the following expressions:
(a) 24× 26= 2 (b) 37× 39 = 3 (c) 36 ÷ 32 = 3 (d) (10 ) (b) 37× 39 = 37 + 9 4 3 = 10 Solution
(a) 24× 26 = 24+6
= 2 10 (c) = 3 16 36 ÷ 32 = 36 − 2 (d) = 34 (10 )
4
3 = 10 4 × 3
= 10 12 Example 2
Simplify each of the following expressions so that it is in the form a n , where n
is a number:
a4× a2
3
6
7
(a) a × a
(b)
(c) ( a 4 )
3
a Solution
(a) a6× a7 = a6+7
= a 13 (b) a4× a2
a4+2
=
a3
a3 = a6
a3 = a6−3
= a3
(c) (a )
4 3...
View
Full
Document
This document was uploaded on 03/03/2014 for the course OLEVEL Mathematic at Beaconhouse School System.
 Fall '13
 Zeeshan

Click to edit the document details