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Indices

 

Indices

an=a×a×a.....n times. Here, ‘a’ is called the base and ‘n’ is called the index or the power or the exponent.

Basic laws of indices:

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Basic formulae for operations on numbers

1.          (a + b)2 = a2 + 2ab + b2;   a2 – b2 = (a + b) (a – b).

2.          (a – b)2=a- 2ab + b

3.          (a + b)2 - (a - b)2 = 4ab

4.          (a + b)2+( a – b )2= 2 (a2+ b2 )

5.          (a + b)(a - b) = a2-b2 

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

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

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 – 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) + 6abc

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

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

6.          n! = 1×2×3×....×(n–2)×(n–1)×n.

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





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