Indices & Surds
GETTING STARTED
Key properties of Indices are as follows:
1 .
eg. 2 -1 = 1/2
2. am x an = am+n
eg. a =3, m=4, n=2.
[For simplicity, we are considering m and n to be integers. They can be any numbers- negative, irrational]
34 x 32 = 81 x 9
= 729
= 36
34 x 32 = 3 (4+2) = 3 6
3 .
eg. a = 6, m = 2, n=3
= 6 -1 = 1/6
4. (am)n = am.n
eg. a = 3, m = 2, n=3
(32)3 = 93 = 729
= 36
= 32×3
Note: a mn and (am)n are not the same.
In a mn power of a is mn ,
whereas for (am)n it is m.n
Eg.
3 23 = 38 whereas
(32)3 = 32 x 32 x 32
= 32+2+2
= 36
In the first case, the power is 2 raised to 3, whereas in the second case is 2 times 3.
5. (a.b)n = an. bn
eg.
Let a =13, b = 12, n =2
(13.12)2 = 132 . 122
= 169 x 144
= 24336
6. ( a⁄ b) n = a n⁄ b n
Eg. a = 6, b =2, n=3
( 6⁄2)3 = 63⁄23
= 216 / 27
= 8
7. a0 = 1
SURDS
We have seen in numbers that the number of form p/q where p and q are integers and q ≠ 0, then p/q is a rational number.
☛ Surds are irrational numbers that are left in root form (√ ) so as to denote the exact value. They are of the form n √a where a is a rational number and n is a positive number.
Eg. √7 , 3√5
☛If bn = a, then n√a = b
n√a denotes nth root of a.
Nth root of a number, a is written as n√a or a1/n
If n is not mentioned, then it is square root of the given number.
Laws of surds:
1. n√a = a1/n
Eg.
√16 = 16 1/2
= 4
2. n√ (a.b) = n√ a x n√ b
eg. 3√ 39 = 3√3 . 3√13
3 .
Eg.
4. (n√a)n = a
Eg. 4√(81)4 = 81
5.
= (a1/n) 1/m
= a1/m.n
eg.
6. (n√a)m = (n√am)
eg. (3√3)4 = (3√81)