Simplification

BODMAS Rule & Series Formulas

Easy-to-understand mathematics notes

1. BODMAS Rule

BODMAS defines the order of operations in mathematics:

• B – Brackets
• O – Of (Powers, Roots)
• D – Division
• M – Multiplication
• A – Addition
• S – Subtraction

Rule: After brackets, always solve from left to right.

2. Types of Brackets

Solve brackets from innermost to outermost:

• Bar / Vinculum (outermost)
• Round Brackets ( )
• Curly Brackets { }
• Square Brackets [ ] (innermost)

3. Important Series Formulas

$$\frac{1}{a b} = \frac{1}{b-a} \left(\frac{1}{a} - \frac{1}{b}\right)$$

$$\frac{1}{a b c} = \frac{1}{c-a} \left(\frac{1}{ab} - \frac{1}{bc}\right)$$

$$\frac{1}{a b c d} = \frac{1}{d-a} \left(\frac{1}{abc} - \frac{1}{bcd}\right)$$

$$\sum_{k=1}^{n} k(k+1) = \frac{n(n+1)(n+2)(n+3)}{4}$$

$$( c(a + b) = ca + cb )$$

Square of Binomial

$$( (a + b)^2 = a^2 + 2ab + b^2 )$$ $$( (a - b)^2 = a^2 - 2ab + b^2 )$$

Product of Sum and Difference

$$( a^2 - b^2 = (a + b)(a - b) )$$

Sum and Difference of Cubes

$$( a^3 - b^3 = (a - b)(a^2 + ab + b^2) )$$ $$( a^3 + b^3 = (a + b)(a^2 - ab + b^2) )$$

Cube of Binomial

$$( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 )$$ $$( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 )$$

Square of Polynomial

$$( (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca )$$

Multiplication of Two Binomials

$$( (x + a)(x + b) = x^2 + (a + b)x + ab )$$

Gauss’s Identity

$$( a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca))$$

Legendre Identity

$$( (a + b)^2 + (a - b)^2 = 2(a^2 + b^2))$$ $$( (a + b)^2 - (a - b)^2 = 4ab )$$ $$( (a + b)^4 - (a - b)^4 = 8ab(a^2 + b^2))$$

Lagrange’s Identity

$$( (a^2 + b^2)(x^2 + y^2) = (ax + by)^2 + (ay - bx)^2)$$ $$((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2 + (ay - bx)^2 + (az - cx)^2 + (bz - cy)^2)$$

4. Solved Examples

Example 1:
1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72

Rewrite as: Σ 1 / k(k+1)
Answer: 8/9

Example 2:
1/6 + 1/24 + 1/60 + ... + 1/1320

Rewrite as: Σ 1 / k(k+1)(k+2)
Answer: 65/264

Example 3 (BODMAS):
5 − [6 − {14 − (18 − 8 + 3)}]

Step 1: (18 − 8 + 3) = 13
Step 2: {14 − 13} = 1
Step 3: [6 − 1] = 5
Step 4: 5 − 5 = 0

Example 4
(√8 x √8)2 + (9)1/2 = (?)3 + 3

(√8 x √8)2 + (9)1/2 = (?)3 + 3
82 + 3 = (?)3 + 3
(?)3 = 64
? = 4