Basics Formula of Algebra

Algebra

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

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

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

4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a+b)3 −3ab(a + b)

5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a−b)3 + 3ab(a − b)

6. a2 − b2 = (a+b)(a − b)

7. a3 − b3 = (a−b)(a2 + ab + b2)

8. a3 + b3 = (a+b)(a2 − ab + b2)

9. an − bn = (a−b)(an−1 + an−2b + an−3b2 + _ _ _ +bn−1)

10. an = a:a:a : : : n times

11. am:an = am+n

12. am

an = am−n if m >n

= 1 if m= n

=

1

an−m if m< n;a 2 R; a 6= 0

13. (am)n = amn = (an)m

14. (ab)n = an:bn

15. _a

b _n

= an

bn

16. a0 = 1 where a 2 R; a 6= 0

17. a−n =

1

an ; an =

1

a−n

18. ap=q = pq ap

19. If am = an and a 6= _1; a 6= 0 then m=n

20. If an = bn where n 6= 0, then a = _b

21. If px;py are quadratic surds and if a + px = py, then a = 0 and x = y

22. If px;py are quadratic surds and if a+px = b+py then a = b and x = y

23. If a;m; n are positive real numbers and a 6= 1, then loga mn = logam+loga n

24. If a;m; n are positive real numbers, a 6= 1, then loga _m

n _= logam−loga n

25. If a and m are positive real numbers, a 6= 1 then logamn = nlogam

26. If a; b and k are positive real numbers, b 6= 1; k 6= 1, then logb a =

logk a

logk b

27. logb a =

1

loga b

where a; b are positive real numbers, a 6= 1; b 6= 1

28. if a;m; n are positive real numbers, a 6= 1 and if logam = logan, then

m=n