LLG Paris-Abu Dhabi Advanced Math and Science Pilot Class
MATHEMATICS Gr10 Lesson 2.
Power of a real number.
De…nition : Set a a real number and n an integer, n
1
a a {z a} an = | n times a
an is called a power n n is called the index of a in an
Examples : 25 = 32 (1:5)3 =
210 = 1024 27 8
23 + 33 = 35
(1:5)2 = 2: 25
( 1:5)3 =
3: 375
1:52 : 1:53 =
( 1:5)2 = 2:25
2: 25
3:375
(2 + 3)3 = 125
Remark 1 : We have to apply carefully the rules of priority (PMDAS)
Remark 2 : For all integer n
1, ( 1)n =
1 when n is even 1 when n is odd
Calculation rules : Rule 1 : Whatever can be a real numbers and n and p whole numbers greater than 1; an
ap = an+p
Proof : Let a be real and nand p two whole numbers, n
1; p
1
an ap = a a} a a} | a {z | a {z n times a p times a = a a} | a {z p + n times a = an+p 1
Rule 2 : Whatever can be a real numbers and n and p whole numbers greater than 1; (an )p = an
p
Proof : Let a be real and n and p two whole numbers, n (an )p =a | =a |
= an
a {z a} n | times a a
a
a |
1; p
a {z a} : : : {z p times a {z a
a
n p
1 a a} | a {z n times a} a
a
a}
p times
Rule 3 : Whatever can be a and b real numbers and n a whole number greater than 1; bn = (ab)n
an
Proof : Let a; b be real and n an whole number, n
=
1
an b n =a a} b| b {z }b | a {z n times a n times b = (a b) (a b) (a b) | {z } n times a b n (a b) = (ab)n
Remark : As a direct consequence of the previous rule, for b 6= 0, a 1 it comes b
a B
n
=
an so : Bn
b=
a and naming B the real 1 b
Whatever can be a and b real numbers with b 6= 0 and n a whole number greater than 1; a b
n
=
an bn
Negative Indices an Let consider p with a 2 R and n and p whole numbers greater than 1: a If n > p,
n
a ap
n times a }| { a a a {z a} p times a = a a} | a {z n p times n p = a z a = a |
2
if n < p,
an ap
n times a }| { a a a {z a} p times a 1 = a} |a a {z p n times 1 = n p a z a = a |
an and …nally, if n = p, p = 1: a So to give the formula of the result a unique form, we decide :
De…nition : For a 6= 0, and n a non 0 integer ,
a
n
=
1 an
and a0 = 1
Now we can write : Rule 4 : Whatever can be a a real number and n and p integers; an = an ap
p
Examples : 2
1
=
1 1 = 21 2
81 = ( 6)3
(2
3 34 3
3)
3
=
=
1 1 = 33 27
34 3 = 23
1
=
1
3 8
25 =2 210
5
=
1 1 = 25 32
Remark 1:
The de…nition is used in science to write many units, for example, the unit of the speed kilometer per hour is written km:h 1 instead of km=h or the concentration unit in chemistry mole per liter is written mol:l 1
Remark 2 : This statements extends the de…nition of powers to the indices in Z . This extension is stable under the operations and the previous proven rules still apply.
3
Rational indices
Going on in the exploration of the calculation rules, p we encounter a new p situation : 2 1 Considering a a strictly positive real number, ( a) = a and a2 = a = a1 so, thinking p 1 to the previous studied rule 2, we decide to write a 2 the number a, and check the validity of the other rules for p this notation: 1 p p 1 1 We have ab = a b so (ab) 2 = a 2 b 2 ; r p 1 a 21 a a a2 = 1 = p so b b b b2 p 1 p n 1 n n a = ( a) so a 2 = (an ) 2 : 1
This choice seems to be valid and give sense to the writing a 2 : We can imagine an analogous p 1 3 de…nition for the cubic root of a positive real number X by X = X 3
(Reminder : The cubic root of a positive number X is the length of a cube for which the area is the given positive number X:)
p There is no such a natural way to de…ne n X for X a positive real number given, but for the time being, we will admit that for each X positive real number, it exists one strictly positive number with a power n equal to X . 1 We will write it X n : According to this new statement, we can de…ne for a 2 R+ and With this de…nition, the previous rules continue to apply.
4
p p p 2 Q, the number ( q a) : q
