2) We know that 3 x 3 x 3 x 3 = 34. Here we say 3 is the base and 4 is the index/ power of 3.
3) We know that 5 x 5 x 5 x 5 x 5 x 5 = 56. Here we say 6 is the base and 6 is the index/ power of 5.
4) We know that 10 x 10 x 10 x 10 x 10 = 105. Here we say 10 is the base and 5 is the index/ power of 10.
Laws of indices :
1) am x an = am+n
2) am / an = am-n
3) (a x b)m = am x b m
4) (a / b)m = am / bm
5) (am)n = amn
6) a0 = 1 ( a is non zero number)
Examples :Rule 1)
1) 23 x 24 = 23+4 = 27; 2) y3 x y8 = y3=8 = y11;
3) 69 x 62 = 69+2 = 611; 4) 53 x 57 = 53+7 = 510.
Examples :Rule 2)
1) 28 / 24 = 28-4 = 24; 2) y13 / y8 = y13-8 = y5.
3) 69 / 62 = 69-2 = 67; 4) 59 / 52 = 59-2 = 52.
Examples :Rule 3)
1) (2 x 5)4 = 24 x 54; 2) ( 7 x 9)2 = 72 x 92;
3) (6 x 17)9 = 69 x 179; 4) (11 x 20)6 = 116 x 206.
Examples :Rule 4)
1) (2 / 5)4 = 24/ 54; 2) ( 7 / 9)2 = 72 / 92;
3) (6 / 17)9 = 69 / 179; 4) (11 / 20)6 = 116 / 206.
Examples :Rule 5)
1) (25)3 = 215; 2) (a3)4 = a12;
3) (102)2 = 104; 4) (x2)-3 = x-6.
Examples :Rule 6)
1) 20 = 1; 2) (2/3)0 = 1;
3) 100000 = 1; 4) x0 = 1.
Fractional Index :
1) 21/2 means the square root of 2.
2) 31/3 means the cube root of 3.
3) 71/4 means the fourth root of 7.
4) 111/5 means the fourth root of 11.
5) a1/n means the nth root of a.
Here if the index is 1/n of a and it is a fraction with n as an non zero integer.
We can find the value of such a number.
For example:
1) 41/2 : Means the square root of 4 = 2
2) 271/3 : Means the cube root of 27 = 3
3) 641/6 : Means the sixth root of 64 = 2
Find the value of the following index number/ Evaluate:
1) 491/2 2) 161/2 3) 25 1/2 4) 1251/3
5) 10001/3 6) 2251/2 7) 5121/3 8) 11/2
Simplify:
1) 181/4 x 183/4 = 181/4+3/4 = 18(1+3)/4 = 181 = 18.
2) 71/3 x 71/2 = 71/3+1/2 = 71/3+1/2= 75/6.
3) p1/2 x p1/2 = p 1/2+1/2 = p1 = p
4) (7 x 9)1/5 = 71/5 x 91/5
5) (41/2)1/9 = 41/18.