# Indices

**Indices**

an=aÃ—aÃ—a.....n times. Here, â€˜aâ€™ is called the base and â€˜nâ€™ is called the index or the power or the exponent.

**Basic laws of indices:**

** **

**Basic formulae for operations on numbers**

1. (a + b)2 = a2 + 2ab + b2; a2 â€“ b2 = (a + b) (a â€“ b).

2. (a â€“ b)2=a- 2ab + b

3. (a + b)2 - (a - b)2 = 4ab

4. (a + b)2+( a â€“ b )2= 2 (a2+ b2 )

5. (a + b)(a - b) = a2-b2

6. (a + b)3 =a3 + 3a2b + 3ab2=a3 +3ab(a + b) + b3

âˆ´ a3 +b3 =(a + b)3 - 3ab (a - b ); or (a + b) (a2+b2- ab)

7. (a - b)3 =a3 + 3a2b + 3ab2-b3 =a3 -3ab(a-b)-b3

âˆ´ a3 - b3 = (a - b)3 + 3ab (a - b); or (a - b)(a2 + b2 + ab)

8. (x + a) (x + b) = x2 + (a + b )x + ab.

9. (x - a) (x + b) = x2 + (b â€“ a ) x - ab.

10. (x - a) (x - b) = x2 - (a + b) x + ab.

11. (a + b + c) 2 = a2 + b2 + c2 + 2ab (ab + bc + ca)

12. (a + b + c)3 = a3 + b3 + c3 + 3ab (a + b) + 3bc (b + c)

+ 3ac(a + c) + 6abcâ€¨

= a3 + b3 + c3 + 3 (a + b) (b + c) (c + a)â€¨= a3 + b3 + c3 + 3( a + b + c) (ab + bc + ac) - 3abc.

13. a3 + b3 + c3 â€“ 3abc = (a + b + c) (a2 + b2 +c2 â€“ ab â€“ bc â€“ ac)

6. n! = 1Ã—2Ã—3Ã—....Ã—(nâ€“2)Ã—(nâ€“1)Ã—n.

(where n is a natural number). 15. The factorial of 0 is 1, i.e. 0! = 1