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# 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 3 = 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) 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

( 62)3 =     6323

= 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)