In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies

Algebra: An Introduction

According to Britannica, Algebra is a branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers. The notion that there exists such a distinct sub-discipline of mathematics, as well as the term algebra to denote it, resulted from a slow historical development. In this article, find a few basic identities and formulae in algebra.

Basic Algebraic Formulas

  • a2 – b2 = (a – b)(a + b)
  • (a+b)2 = a2 + 2ab + b2
  • a2 + b2 = (a – b)2 + 2ab
  • (a – b)2 = a2 – 2ab + b2
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
  • (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
  • (a – b)3 = a3 – 3a2b + 3ab2 – b3
  • a3 – b3 = (a – b)(a2 + ab + b2)
  • a3 + b3 = (a + b)(a2 – ab + b2)
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3
  • (a – b)3 = a3 – 3a2b + 3ab2 – b3
  • (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)
  • (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)
  • a4 – b4 = (a – b)(a + b)(a2 + b2)
  • a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)

 

  • If n is a natural number, an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
  • If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
  • If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
  • (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….
  • Laws of Exponents(am)(an) = am+n
    (ab)m = ambm
    (am)n = amn
  • Fractional Exponents
    a0 = 1
    am/an=amn
    am = 1/am
    am = 1/am

Solved Equations

Question: If x3 + y3 = 9 and x + y = 3, then the value of x4+y4 is
Solution:
x3+y3 = (x + y) × (x2 − xy + y2)

Putting given values of x3+y3 and (x + y)
9 = 3 × ((x+y)2 − 3xy)
= 3 × (9 − 3xy)
= 27 − 9xy

9xy = 18
xy = 2

x4 + y4 = (x2 + y2)2 – 2x2y2
= (x2 + y2)2 – 2*4 . . . . . [Putting value of xy]
= ((x + y)2 – 2xy)2 – 2*4 . . . . . [Putting values of (x+y) and xy]
= (9 – 4)2 – 2*4
= 17

Question 2: 43 × 42 = ?
Solution:
Using the exponential formula (am)(an) = am+n
where a = 4
43 × 42
= 43+2
= 45
= 1024

Question 3: Find the value of 42 – 32
Solution: Using the formula a2 – b2 = (a – b)(a + b)
where a = 4 and b = 3
(a – b)(a + b)
= (4 – 3)(4 + 3)
= 1 × 7
= 7
For a similar article on geometry, click here!

Shock your Dad with more marks than he expected.

Access 300,000+ questions curated by India’s top rankers.

+91
No thanks.

Request a Free 60 minute counselling session at your home

Please enter a valid phone number
  • Happy Students

    7,829,648

    Happy Students
  • Questions Attempted

    358,177,393

    Questions Attempted
  • Tests

    3,028,498

    Tests Taken
  • Doubts Answered

    3,020,367

    Doubts Answered