Study Materials

Simplifications

BODMAS Rule

Details of BODMAS Rule are given below:

B	Brackets first
O	Orders (ie Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

Note:

• Order of the letter which is used in BODMAS Rule is always fixed.
• Absence of any operation or more than one operations does not change the order of BODMAS.
• ‘Of’ means multiplication.

BRACKETS
When all brackets are present in a question, in that condition ViCiCuSq Rule is applied.nDetails are given below:
ADDITION
Sum of consecutive n-natural numbers =n(n+1)/2
Sum of Xnth Terms=X.n(n+1)/2
Sum of Squares of Consecutive n-natural numbers=n(n+1)(2n+1)/6
Sum of cubes=[n(n+1)/2]2
Sum of even numbers=X(X+1) where X=n/2

ALGEBRAIC FORMUL

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 time

Powers and Root,Square,Cube,Indices,Surds Squaring

1. Short Methods in Squaring
    Let a,b denote numbers
   a²=a²-b²+b²=(a²-b²)+b²
   or a²=[(a+b)(a-b)]+b²
   (a+b)²=a²+b²+2ab
   (a-b)²=a²+b²-2ab
   Example: Find (1213)²=?
   =[(1213-13)(1213+13)]+13²
   =(1200*1226)+169
   =1471369
    
   
   
   Squaring of a number ending in 5
   Multiply the number formed after deleting 5 at the unit’s place with the number one higher than it, Annex 25 on the right side of the product and you will get the square of the given number.
   Example: Find 165²=?
   16*17=272
   So, 165²=27225
   Square of Decimal Number
   Find the square of the number ignoring the decimal point. Put the decimal point leaving double the number of digits(from the right) as compared to that in the given number. In other words,the position of decimal place in the square is double of that in the original number.
   Example: Find the square of 14.52=?
   (14.52)²=210.8304
   Square of Fraction
   Square of 1½,2½,3½,4½ etc.
   Multiply the integral part by one more than it. Add ¼ to the product and you will get the square of the given half fraction.
   Square of number consisting of 9′s only
   Let the number consists of n 9′s.
   Write down (n-1) 9′s, followed by one 8, then (n-1) zeros and finally annex 1 at the end.
   Example: Find (99999)²=?
   (99999)²= 9999800001
   SQUAREROOTS
   Method 1: By Factorization
   This method is generally used where the given number is perfect square or when the number can be written as product of such factors whose square roots are known.
   In factorization method we write the given number as product of prime factors and take the product of the prime factors, choosing one out of every pair.
   Method 2: By Division
   Step 1: Mark off groups of two digits starting from right. The extreme group may be either single digit or a pair.
   Step 2: Start division process from the extreme left group
   Step 3: For the second stage add the quotient to the divisor. The divisor of this stage will be equal to this sum with the quotient for this stage suffixed to it. The next dividend is always obtained by annexing the next pair of digits (of the dividend) to the remainder.
   Step 4: For the next stage again add the divisor and the quotient of the previous stage. The divisor for this stage will be formed in the same manner as explained for the second stage in Step 3.
   Step 5: Continue step 4 till all the groups get exhausted, in case a remainder is left, annex two 0′s to it and put a decimal point in the quotient.
   At every stage after this we will annex two 0′s to the remainder. Continue to the number of decimal places required in the result. The quotient is equal to the square root of the given number.