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# Basic Rules of Simplification

### BODMAS Rule

It defines the correct sequence in which operations are to be performed in a given mathematical expression to find the correct value. This means that to simplify an expression, the following order must be followed –
= Bracket,
= Order (Powers, Square Roots, etc.)
D = Division
= Multiplication
= Subtraction

Easiest way to choose simplification questions:

STEP 1: Know about BODMAS Rule. Following are the list of priority given for brackets and signs.

STEP 2: If an expression Contains brackets, the expression within thebrackets should be simplified first.

STEP 3: If an expression contains ‘Of’, multiplication, division, addition and subtraction, thenof should be performed first then followed by multiplication or division.

Proceeding from left to right, addition and subtraction are carried out in the order in which the sign of addition and subtraction are given.

If expression contains‘Of’ and Division – always do ‘Of’ and then do division

STEP 4: If expression involves all thefouroperations,then multiplication and divisionis carried outfirst in the order in which they are given from left to right. The same rules are carried out for addition and subtraction

### Important Parts of Simplification

• Number System
• HCF & LCM
• Square & Cube
• Fractions & Decimals
• Surds & Indices

### Number System

• Classification
• Divisibility Test
• Division& Remainder Rules
• Sum Rules

#### Classification

Types Description
Natural Numbers:
all counting numbers ( 1,2,3,4,5….∞)
Whole Numbers:
natural number + zero( 0,1,2,3,4,5…∞)
Integers:
All whole numbers including Negative number + Positive number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞)
Even & Odd Numbers :
All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞)
Prime Numbers:
It can be positive or negative except 1, if the number is not divisible by any number except the number itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞)
Composite Numbers:
Natural numbers which are not prime
Co-Prime:
Two natural number a and b are said to be co-prime if their HCF is 1.

#### Divisibility

Numbers IF A Number Examples
Divisible by 2 End with 0,2,4,6,8 are divisible by 2 254,326,3546,4718 all are divisible by 2
Divisible by 3 Sum of its digits  is divisible by 3 375,4251,78123 all are divisible by 3.  [549=5+4+9][5+4+9=18]18 is divisible by 3  hence 549 is divisible by 3.
Divisible by 4 Last two digit divisible by 4 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4.
Divisible by 5 Ends with 0 or 5 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5.
Divisible by 6 Divides by Both 2 & 3 4536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6.
Divisible by 8 Last 3 digit divide by 8 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8.
Divisible by 10 End with 0 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10.
Divisible by 11 [Sum of its digit in
odd places-Sum of its digits
in even places]= 0 or multiple of 11
Consider the number 39798847
(Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3)
(23-12)

23-12=11, which is divisible by 11. So 39798847 is divisible by 11.

#### Division & Remainder Rules

Suppose we divide 45 by 6
hence ,represent it as:
dividend = ( divisor✘quotient ) + remainder
or
divisior= [(dividend)-(remainder] / quotient
could be write it as
x = kq + r where (x = dividend,k = divisor,q = quotient,r = remainder)
Example:
On dividing a certain number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder ?
Number = 342k + 47
( 18 ✘19k ) + ( 18 ✘2 ) + 11
18 ✘( 19k + 2 ) +11.
Remainder = 11

#### Sum Rules

(1+2+3+………+n) = 1/n(n+1)
(12+22+32+………+n2) = 1/n (n+1) (2n+1)
(13+23+33+………+n3) = 1/4 n2 (n+1)2

Arithmetic Progression (A.P.)

a, a + d, a + 2d, a + 3d, ….are said to be in A.P. in which first term = a and common difference = d.
Let the nth term be tn and last term = l, then

a) nth term = a + ( n – 1 ) d
b) Sum of n terms = n/2 [2a + (n-1)d] c) Sum of n terms = n/2 (a+l) where l is the last term

### H.C.F. & L.C.M.

• Factorization & Division Method
• HCF & LCM of Fractions & Decimal Fractions

#### Methods

On Basis
H.C.F. or G.C.M
L.C.M.
Factorization Method
Write  each number as the product of the prime factors. The product of least powers of common prime factors gives H.C.F.
Example:
Find the H.C.F. of 108, 288 and 360.

108 = 22✘33, 288 = 25✘32 and 360 = 23✘5✘32

H.C.F. = 22✘32=36

Write each numbers into a product
of prime factors. Then, L.C.M is
the product of highest powers of
all the factors.
Examples:
Find the L.C.M. of 72, 108 and 2100.
72=23✘32,108=33✘22,
2100=22✘52✘3✘7.
L.C.M.=23✘33✘52✘7=37800
Division Method

Let we have two numbers .Pick the smaller one and divide it by the larger one. After that divide the divisor with the remainder. This process of dividing the preceding number by the remainder will repeated until we got the zero  as remainder.The last divisor is the required H.C.F.
Example:

H.C.F. of given numbers = 69
Let we have set of numbers.
First of all find the number
which divide at least two of
the number in a given set of
number.remainder and
not divisible numbers
will carry forward as it is.
Repeat the process till
at least  two number is
not divisible by any number
except 1.The product of
the divisor and the
undivided numbers is the required
L.C.M.

Example:
Find the L.C.M. of 12,36,48,72

H.C.F. & L.C.M. of Fractions
H.C.F. =  H.C.F. of Numerator / L.C.M. of Denominators L.C.M. = L.C.M. of Numerator / H.C.F. of Denominators
Product of H.C.F. & L.C.M.
H.C.F * L.C.M. = product of two numbers
Decimal numbers H.C.F. of Decimal numbers
Step 1. Find the HCF of the given
numbers without decimal.
Step 2.Put the decimal point ( in the
HCF of Step 1) from right to left
according to the MAXIMUM
deciaml places among the given numbers.
L.C.M. of Decimal numbers
Step 1. Find the LCM of the given
numbers without decimal.
Step 2.Put the decimal point ( in the
LCM of Step 1) from right to left
according to the MINIMUM
decimal places among the given numbers.

### Square & Cube

• Square & Cube
• Square Root & Cube Root
• Factorization Method
 Perfect Square Non-Perfect Square last digit is 1, 4, 9, 6, 5 last digit is 2, 3, 7, 8

Square Root & Cube Root

### Fractions & Decimals

On Basis Explanation
Decimal Fractions
A number with a denominator of power of 10 is a decimal fractions.
1/10= 1 tenth; 1/100= 0.1;38/100=0.38
Vulgar Fractions
Conversion of 0.64(decimal number) into a Vulgar Fraction.First of all write the numeric digit 1 in the denominator of a number (like here 0.64) and add as many numeric zeros as the digit in the number after decimal point.After that removes the decimal point from the given number.At last step just reduce the fraction to its lowest terms. So, 0.64 = 64/100=16/25;25.025 = 25025/1000 = 1001/4
To perform the addition and subtraction of a decimal fraction could be done through placing them right under each other that the decimal points lie in one column.
3.424+3.28+.4036+6.2+.8+4
3. 424
3. 28
. 4036
6. 2
. 8
+4______
18. 1076Multiplication of a Decimal Fraction
To find the multiplication of decimal fraction , first of all you need to remove the decimal point from the given numbers and then perform the multiplication after that assign the decimal point as many places after the number as the sum of the number of the decimal places in the given number.
Step 1. 0.06*0.3*0.40
Step 2. 6*3*40=720
Step 3. 0.00720
Multiplication of a decimal fraction by power of 10
A multiplication of a decimal fraction by a power of 10 can be perform through shifting the decimal point towards right as many places as is the power of 10.
like 45.6288*100=45628.8, 0.00452*100=0.452
Division

Comparison of Fractions To compare the set of fractions numbers,first of all you need to convert each fraction number or value into a equal decimal value and then it will be became easy for you to assign them ( the numbers or value) in a particular way( ascending or descending order).
3/5,4/7,8/9 and 9/11 Arranging in Ascending Order
3/5= 0.6, 4/7 = 0.571, 8/9 = 0.88, 9/11 = 0.818.
Now, 0.88 > 0.818 > 0.6 > 0.571
8/9>9/11>3/5>4/7
Recurring Decimal Recurring Decimal
A decimal number in which after a decimal point a number or set of number are repeated again and again are called recurring decimal numbers.It can be written in shorten form by placing a bar or line above the numbers which has repeated.

Pure Recurring Decimal
A decimal number in which all digits are repeated after a decimal point.

Mixed Recurring Decimal
A decimal number in which certain digits are repeated only.

### Surds & Indices

• Some Rules of Indices
• Some Rules of Surds

Learnsquares and cubes of number:

Squares(12 to 302):

• 1– 1
• 2– 4
• 3– 9
• 4– 16
• 5– 25
• 62– 36
• 72– 49
• 82– 64
• 92– 81
• 102-100
• 11–121
• 122-144
• 132– 169
• 142– 196
• 152– 225
• 162– 256
• 172– 289
• 182– 324
• 192– 361
• 202– 400
• 212– 441
• 222– 484
• 232– 529
• 242– 576
• 252– 625
• 262– 676
• 272– 729
• 282– 784
• 292– 841
• 302– 900

Cubes (13to 153):

• 13– 1
• 23– 8
• 33– 27
• 43– 64
• 53– 125
• 63– 216
• 73– 343
• 83– 512
• 93– 729
• 103– 1000
• 113– 1331
• 123– 1728
• 133– 2197
• 143– 2477
• 153– 3375

Fast and Easy Method to Take Square Root- Math Tricks:

Fast and Easy Method to Take Square Root is given here, candidates those who are preparing for banking and all other competitive exams can use this trick in the examination to save time. We have given the link to download in PDF.

POINTS TO REMEMBER:

• When 22= 4, then √4 = 2
• Here 4 is thesquareof 2
• 2 is thesquare rootof 4
• ASquareof a number cannever end with2, 3, 7and 8

Table 1:

 One’s digit of a square One’s digit of the square root 1 1 or 9 4 2 or 8 5 5 6 4 or 6 9 7 or 3

To find the square of a number which is amultiple of ‘5’

252 = [2×3] 52

= [6] 25 = 625

i.e., AB2whereB=5

AB2= [A× next number] B2

For example, 852= [8×9] 25 =7225

1152= [11×12] 25 =13225

1552= [15×16] 25 =24025

This method can be followed for all numbers divisible by 5

TYPE 1:

To find the square root of a 3-digit number

EXAMPLE:√841

STEP 1: Consider the one’s digit of the given number i.e., 1

From Table 1, if the one’s digit of the square is ‘1’ then the square root would either end with‘1’ or‘9’

STEP 2:Alwaysignore the ten’s digit of the given number

STEP 3:Now the remaining number other than the one’s and the ten’s digit in the given number is‘8’

Consider asquare-root of a square which isnearer to as well aslesser than ‘8’.

Here it is ‘4’ which is nearer to as well as lesser than ‘8’. Hence the square root of 4 i.e.,‘2’is taken

STEP 4: we already know the one’s digit of the square root to be either 1 or 9 from STEP 1

Therefore the square root of ‘841’ lies between21and29

STEP 5:

Take a numberdivisible by ‘5’ between 21 and 29, that is‘25’

252= [2×3] 25 =625

Now625< 841

252is itself lesser than 841. Then212will bemuch lesser than841.

Therefore, the remaining option is ‘29’

√841 = 29

TYPE 2:

To find the square root of a 4-digit number

EXAMPLE:√8464

STEP 1: Consider the one’s digit of the given number i.e., 4

From Table 1, if the one’s digit of the square is ‘4’ then the square root would either end with‘2’ or‘8’

STEP 2:Alwaysignore the ten’s digit of the given number

STEP 3:Now the remaining numbers other than the one’s and the ten’s digit in the given number is‘84’

Consider asquare-root of a square which isnearer to as well aslesser than ‘84’.

Here it is‘81’ which is nearer to as well as lesser than ‘84’. Hence the square root of 81 i.e.,‘9’ is taken

STEP 4: we already know the one’s digit of the square root to be either 2 or 8 from STEP 1

Therefore the square root of ‘8464’ lies between92and98

STEP 5:

Take a numberdivisible by ‘5’ between 92 and 98, that is‘95’

952= [9×10] 25 =9025

Now9025> 8464

952is itself greater than 8464. Then982will bemuch greater than8464

Therefore, the remaining option is ‘92’

√8464 = 92

TYPE 4:

To find the square root of a 5-digit number

EXAMPLE:√18769

STEP 1: Consider the one’s digit of the given number i.e., 9

From Table 1, if the one’s digit of the square is ‘9’ then the square root would either end with‘3’ or‘7’

STEP 2:Alwaysignore the ten’s digit of the given number

STEP 3:Now the remaining numbers other than the one’s and the ten’s digit in the given number is‘187’

Consider asquare-root of a square which isnearer to as well aslesser than ‘187’

Here it is‘169’ which is nearer to as well as lesser than 187. Hence the square root of 169 i.e.,‘13’ is taken

STEP 4: we already know the one’s digit of the square root to be either 3 or 7 from STEP 1

Therefore the square root of ‘18769’ lies between133and137

STEP 5:

Take a numberdivisible by ‘5’ between 133 and 137, that is‘135’

1352= [13×14] 25 =18225

Now18225< 18769

1352is itself smaller than 18769. Then1332will bemuch lesser than18769

Therefore, the remaining option is ‘137’

√18769 = 137

Shortcut Tricks to find the Square of Numbers-

Simple Shortcut Tricks to find the Square of Numbers is given here, which was very helpful for the speedy calculations. Candidates those who are preparing for banking and all other competitive exams can use also download this in PDF.

SQUARE OF NUMBERS

• Square of a number is the product obtained by multiplying a number by itself.

2×2 = 4

11×11 = 121

• To find the square of two digit numbers(10-99)we can consider the following steps.

1)   Let the 2-digit number be = AB

2)   Now to find AB2

3)   Unit digit of square = B2

4)   Ten’s digit of square = 2×A×B (+ Carry if any from the previous step)

5)   The rest of the digits of square = A2(+ Carry if any from the previous step)

FOR EXAMPLE

To find of Square of 67,

672 = ?

AB2 = 672

STEP 1:

B2=72=49

From Step 1, here “9” is the unit digit and “4” is carry

à 672= _ _ _ 9

STEP 2:

[2×A×B]+ Carry from previous step i.e.,4

2×A×B = 2×6×7 = 84

Add Carry ‘4’ with the above ‘84’ we get [84+4=88]

Therefore at the end of the 2nd Step

Here “8” is the unit digit and “8” is carry

à672= _ _ 8 9

STEP 3:

A2= 62 = 36

There is a carry of ‘8’ from the previous step

Therefore, [36+8] = 44

• To find the square of a number, which is amultiple of ‘5’

AB2= [A× next number] B2

252 = [2×3] 52

= [6] 25 = 625

i.e., AB2whereB=5

AB2= [A× next number] B2

For example, 852= [8×9] 25 =7225

1152= [11×12] 25 =13225

1552= [15×16] 25 =24025

This method can be followed for all numbers divisible by 5

Fast and Easy Method to Take Cube Root- Math Tricks:

Fast and Easy Method to Take Cube Root is given here, candidates those who are preparing for banking and all other competitive exams can use this trick in the examination to save time.

POINTS TO REMEMBER:

• When 23= 8, then (8)1/3= 2
• Here 8 is thecubeof 2
• 2 is thecube rootof 8

Table 1:

 One’s digit of a cube One’s digit of the cube root 1 1 2 8 3 7 4 4 5 5 6 6 7 3 8 2 9 9 0 0

TYPE 1:

To find the cube root of a 4-digit number

EXAMPLE:(9261)1/3

STEP 1: Consider the one’s digit of the given number i.e.,1

From Table 1, if the one’s digit of the cube is ‘1’ then the cube root would also end with‘1’

STEP 2:Alwaysignore the ten’sand the hundredth digits of the given number i.e., ignore‘2’ and‘6’ in the given number

STEP 3:Now the remaining number other than the one’s, ten’s and the hundredth digit in the given number is‘9’

Consider acube-root of a cube which isnearer to as well aslesser than ‘9’.

Here it is ‘8’ which is nearer to as well as lesser than ‘9’. Hence the cube root of 8 i.e.,‘2’ is taken

STEP 4: we already know the one’s digit of the cube root is1 from STEP 1

Therefore the cube root of ‘9261’ is21

Therefore(9261)1/3 = 21

TYPE 2:

To find the cube root of a 5-digit number

EXAMPLE:(32768)1/3

STEP 1: Consider the one’s digit of the given number i.e., 8

From Table 1, if the one’s digit of the cube is ‘8’ then the cube root would end with‘2’

STEP 2:Alwaysignore the ten’s and the hundredth digitsof the given number i.e., ‘6’ and ‘7’ here

STEP 3:Now the remaining numbers other than the one’s, ten’s and the hundredth digits in the given number is‘32’

Consider acube-root of a cube which isnearer to as well aslesser than ‘32’.

Here it is‘27’ which is nearer to as well as lesser than ‘32’. Hence the cube root of 27 i.e.,‘3’ is taken

STEP 4: we already know the one’s digit of the cube root to be2 from STEP 1

Therefore the cube root of ‘32768’ is32

Therefore(32768)1/3 = 32

Example 1: 212 / 49 × 6

Solution:From the above question if we know the square value of 212, then this question will be easily solved

STEP 1:212= 441

STEP 2:441/49= 9

STEP 3:9×6 = 54

STEP 4:Hence the answer for above series is54

2) REMEMBER FREQUENTLY ASKED FRACTION VALUES

• 5% = 0.05
• 6 ¼ % = 0.0625
• 10% = 0.1
• 12 ½ = 0.125
• 16 × (2/3)% = 0.166
• 20 % = 0.2
• 25 % = 0.25
• 33 × (1/3)%= 0.33
• 40 % = 0.4
• 50% = 0.5
• 60% = 0.6
• 66 × (2/3) =0.66
• 75 %= 0.75
• 80 %= 0.8
• 90 % = 0.9
• 100% = 1
• 125 % = 1.25
• 150% = 1.5
• 200 % = 2
• 250 % =2.5

EXAMPLE 2: 60% of 250 +25% of 600

STEP 1: Know the values of 60% =0.6 and 25 % = 0.25

STEP 2: Now directly multiply 0.6×250 + 0.25×600

STEP 3:0.6×250= 150

0.25×600=150

STEP 4: 150+ 150 = 300

STEP 5:Hence the answer for above series is300

3) Solve mixed fraction – Multiplication

EXAMPLE 3: 2×(3/5) × 8×(1/3) + 7 ½ × 2×(2/3)

STEP 1:2×(3/5) × 8×(1/3) = (13/5) × (25/3) = 65/3

STEP 2:  + 7 ½ ×2×(2/3)= 43/6 × 12/5 = 86/5

STEP 3:65/3 + 86/5 = 38×(15/13)

STEP 4:hence the answer for above series is38×(15/13)

Example 4:19×(3/5) + 23×(2/3) – 24×(1/5)

STEP 1:Take all the whole number outside the bracket i.e. 19+23 -24 = 18

STEP 2:Add fractions within bracket 18×[(3/5) + (2/3) – (1/5)] = 18(16/15)

STEP 3:  Hence the answer for above series is18(16/15)

Example 5: (?)2+18×12= 62×5×2

STEP 1:Multiply 18 × 12 = 216

STEP 2:Square of 6 = 36

STEP 3:Multiply 36 ×5×2= 360

STEP 4:(X)+216 = 360

STEP 5:(X)2  = 360-216 = 144

STEP 6:Therefore X = 12

## Multiplication Tricks – Find solution within 20 seconds

Today I am going to share quick multiplication tricks that will help you find the solution within 20 seconds. Must read – 10 Coolest Maths tricks

### Multiply by 9,99,999,etc…

56*99=5544
Step 1:Place a zero at the end for each 9 :5600
Step 2 : Subtract the original number from Step 1 like this 5600-56=5544

### Multiply by 125

68*125=8500
Step 1 :Place three zeros at the end of the number :68000
Step 2: Divide the number from Step 1  by 8:68000/8=8500
64*125 is the same as :
32*250 is the same as
16*500 is the same as
8*1000
64*125

Step 1. Each time you just need to pick 125 multiply it by 8 will get 1000

Step 2. Pick 64 and divide it by 8 will get 8
Step 3. Multiply the results with each other 8* 1000
Hence Solution is 8000
[Hint: Just remember 125*8=1000]

### Multiply two digits numbers ending in 1

51*31=1581
Step 1: Multiply the left most digits : 5*3=15
Step 2: Add the left most digits:5+3=8
Step 3: Places the result from Step 2 next to the result from Step 1:158
Step 4 : Places 1 next to the result from  Step 3 : 1581

### Multiply numbers between 11 and 19

14*18=252
Step 1: Add the larger  number to the right most digit of the other number:18+4=22
Step 2: Put a 0 at the end of the result from step 1:220
Step 3 :Multiply the right most digits of both original numbers : 8*4=32
Step 4:Add Step 2 and step 3 :220+32=252

### Multiply two digit number by 11

53*11=583
Step 1: Add the both digts of the two digit number:5+3=8
Step 2: Place the result in between both digits : 583
59*11=649
Step 1: 5+9=14
Step 2 : Carry the 1 when the result is greater than 9:5+1=6

Step 3: 649

### Multiply by 5

1234 *5 =6170
Step 1 : Divide  the number by 2 :1234/2=617

Step 2: Multiply  the result from Step 1 by 10 : 617*10=6170

### Multiply by 25

18*25=450
Step 1: Divide  the number by 4:18/4
Step 2: Multiply  the number from Step 1 by 100: 4.5 * 100 = 450

### Multiply by 9

56*9=504
Step 1: Multiply the number by 10: 56*10=560
Step 2: Subtract the original number from Step 1: 560-56=504

### Factorization

By Factoring number,you can break down problems into simpler multiplication tasks.Also,you may be able to apply some techniques you learned.
21*33
step 1 : 21*11*3
Step 2: 231*3
Step 3 :693
67*81
Step 1: 67*9*9
Step 2:603*9=5427
28*125=3500
Step 1: 28*125
Step 2: 28*25*5
Step 3:28*(100/4)*5
Step 4:28/4*100*5

Step 5:7*500=3500

### Some Special type

#### When sum of unit digit is 10 and remaining digit is same.

43×47 = 4 × ( 4+1 ) / 3×7

= 4×5/21
20/21
Ans = 202172×78 = 7×8 / 2×8
56/16
Ans = 5616104 × 106 = 10× ( 10+1 ) / 4×6
= 10 × 11/24
110/24
Ans = 11024

#### When sum of ten’s digit is 10 and unit digit is same

46 × 66
= ( 4×6 ) +6 / 6×6
= 24 + 6/36
30/ 36
Ans = 303683 × 23
= ( 8×2 ) +3 / 3×3
19/09
Ans = 190992 × 12
= ( 9×1 ) + 2/2 × 2
11/04
Ans = 1104

#### When unit digit is 5 in both the numbers and difference between both number is 10.

75 × 65

=  6 × ( 7+1 ) / 75
=  48/75
Ans = 4875

45 × 35
= 3 × ( 4+1 )/75
= 15 /75
Ans = 1575

105 × 95
= 9 × ( 10+1 ) / 75
= 99/75
Ans = 9975

### Multiply by 11,111,1111….so on

Ques 1. 111111111 ✘ 111111111 = ?

Sol:
No of digits in multiplier = 9
Write in ascending order from left side like this:
987654321
and now 9-1=8
write it in descending order just after it
12345678
now you will get like this:
12345678987654321
hence
111111111✘111111111 = 12345678987654321

Ques 2. 1111111111 ✘ 1111111111 = ?
Sol:
No of digits in multiplier =10
Write in ascending order from left side like this:
10 9 8 7 6 5 4 3 2 1
and now 10-1=9
write it in descending order just after it
1 2 3 4 5 6 7 8 9
and after it just add the carry
1 2 3 4 5 6 7 8/ 9/ 10 9 8 7 6 5 4 3 2 1
8+1/   9+1 / 0
1 2 3 4 5 6 7 9 0 0 9 8 7 6 5 4 3 2 1
now you will get like this:
1234567900987654321
hence
1111111111✘1111111111 = 1234567900987654321

Ques 3. 1111111✘2222222 = ?
Sol:
No of digit in the multiplier is 7 then let n=7;
Now Just multiply the digit 2 from 1 to 7 time & arrange them from extreme left to right in ascending order,you will get like this:
14 12 10 8 6 4 2
and now just subtract one from n.like this n=7,so n-1=6.
Multiply the digit 2 from 1 to 6 time & arrange them from just right after it,you will get like this:
2 4 6 8 10 12
Now placing both outcome like this & add the carry
2 4 6 8 10 12 14 12 10 8 6 4 2
8+1/0+1/2+1/4+1/2+1
2 4 6 9 1 3 5 3 0 8 6 4 2

Ques 4. 1111111✘5555555 = ?
Sol:
No of digit = 7
Now Just multiply the digit 5 from 1 to 7 time & arrange them from extreme left to right in ascending order,you will get like this:
35 30 25 20 15 10 5
Just right after it perform same action but in descending order & till 6 times only.like this:
5 10 15 20 25 30
Now placing together ,just add the carry
5 10 15 20 25 30 35 30 25 20 15 10 5
6 1 7 2 8 3 8 2 7 1 6 0 5
1111111✘5555555=6172838271605

### # Division Method

• Step 1. Make Pair of digits of given number from left to right
• Step 2. Pick first pair, like here 6 find the square which is equals to 6 or less than it.Like 2
• Step 3. So Place it to in the section of Quotient as well as in the divisor.
• Step 4: then  subtract from square of no which is equals to 6 or less than it with 6
• Step 5. Now comes to second pair bring it down like here 40 ,double the quotient like 2 = 4 and write the result on the left of 240 .It is just like division.Now repeat From Step 2 until you got the remainder zero.

### # Square Root of a Decimal Fraction

• Step 1. Make the pair of integer part first.
• Step 2. Now find whether the decimal part is odd or even if it is odd then make it odd by placing at the end of it zero.
• Now just find the square root by the division method as discussed above and don’t forget to put the decimal point in the square root as the integer part is over.

### # Method of Finding Cube Root of Perfect Cube

Fast Math and Shortcut Rules for Multiplication:

Useful Shortcut rules for Multiplication which was more helpful in the fast math to solve the aptitude questions were given below. Candidates those who are preparing for the banking and all competitive exams can use this

Type-1: If the unit figure is same and the sum of the tens figure is 10, then follow the below method.

General Shortcut Method:

[Tens fig. × Tens fig. + Unit fig.] [Unit fig × Unit fig]

Example:

86 × 26 = [8 × 2 + 6] [6 × 6] = [22] [36]; so answer is: 2236.

Note: Here, the unit figure denotes the number that present in the ones digit (6, 6), and tens figure denotes the number that present in the tens digit (8, 2).

Type-2: If the sum of the unit figure is 5 and the tens figure are equal. Then follow the below method.

General Shortcut Method:

[(Tens figure)2+ ½ × Tens figure] [Unit fig. × Unit fig]

Example:

83 × 82 = [82+ ½ × 8] [ 3 × 2 ] = [68] [06] So the answer is: 6806

Type-3: If the unit figures are same and the sum of tens figures is 5.

General Shortcut Method:

[Tens fig × Tens fig + ½ × Unit fig] [(Unit fig.)2]

Example:

36 × 26 = [3 × 2 + ½ × 6] [62]

= [9] [36]; Answer is 936.

Type-4: If the unit figures are 5 and difference between the tens figures is 1 then the rule is,

General Shortcut Method:

[(Larger tens fig + 1) × (Smaller tens fig)] [75]

Example:

35 × 45 = [(4 + 1) × 3] [75]

= [15] [75]; So the Answer is, 1575.

More Shortcuts Will be Updated Soon.

Fast Math and Shortcut Rules for Division:

Useful Shortcut rules for Division which was more helpful in the fast math to solve the aptitude questions were given below. Candidates those who are preparing for the banking and all competitive exams can use this.

1.)DIVISIBLE BY 2:

A number will be divisible by 2, if the unit digit in the number is 0, 2, 4, 6 and 8.

Example: Numbers like, 56456, 32658, 89846 are divisible by 2.

2.)DIVISIBLE BY 4:

A number will be divisible by 4, if the last two digits of the number is divisible by 4.

Example: Numbers like 56536 is divisible by 4, because the last two digits of this number is divisible by 4 and the number 546642 is not divisible by 4 because the last two digits of this number is not divisible by 4.

3.)DIVISIBLE BY 6:

A number will be divisible by 6, if that number is divisible by both 2 and 3.

Example: 36 is divisible by 6 because 36 is divisible by both 2 and 3.

4.)DIVISIBLE BY 8:

A number will be divisible by 8, if the last three digits of that number are divisible by 8.

Example: 565144 is divisible by 8 because the last three digits 144 is divisible by 8. And the number 554314 is not divisible by 8 because the last three digits 314 is not divisible by 8.

5.)DIVISIBLE BY 5:

A number will be divisible by 5 if the unit digit is either 0 or 5.

Example: Numbers like 565520 and 898935 are divisible by 5.

6.)DIVISIBLE BY 3:

A number will be divisible by 3, if the sum of the digits in the number is divisible by 3.

Example: 658452 is divisible by 3 because the sum of the numbers is divisible by 3, 6+5+8+4+5+2= 30, which is divisible by 3.

The number 456455 is not divisible by 3, because the sum of the number is not divisible by 3. 4+5+6+4+5+5= 29 this is not divisible by 29.

7.)DIVISIBLE BY 9:

A number will be divisible by 9, if the sum of the digits in the number is divisible by 9.

Example: 898686 is divisible by 9 because the sum of the numbers is divisible by 9,

8+9+8+6+8+6= 45, this is divisible by 9.

8.)DIVISIBLE BY 11:

A number will be divisible by 11, if the difference of the sum of the digits in the Odd places and Sum of the digits in the Even places, is either zero or divisible by 11.

Example: 502678 is divisible by 11 because, the sum of the digits of the odd places, 5+2+7= 14, sum of the digits in the even places, 0+6+8=14, the difference is 14-14=0, so this number is divisible by 11.

9.)DIVISIBLE BY 12:

A number is divisible by 12, if the number is divisible by both 3 and 4.

Example: 144 is divisible by 12, because it is divisible by both 3 and 4.

10.)DIVISIBLE BY 10:

Any number that ends with zero will be divisible by 10.

Fast Math and Shortcut Rules for Subtraction- Rule-I:

Useful Shortcut rules for Subtraction which was more helpful in the fast math to solve the aptitude questions were given below. Candidates those who are preparing for the banking and all competitive exams can use this.

Rule-I:Borrowing and Paying Back Method:

This method is the quickest method of subtraction. This method is also called equal additions method.

Example (1): Suppose we have to subtract 55 from 91. Mentally we have to increase the number to be subtracted to the nearest multiple of 10 i.e., increase 55 to 60 by adding 5 to it. Mentally increase the other quantity by the same amount i.e., by 5. Therefore, the problem is 96 minus 60 i.e., our answer is 96 – 60= 36.

Example (2): Sometimes it is useful to increase the number to be subtracted to the nearest multiple of 100 for example 442 – 179. Therefore 179 becomes 200 by adding 21 and 442 becomes 463 by adding 21. Then the problem becomes 463 – 200= 263. Now we see that 463 – 200 is easier than 442 – 179. The result is same as 263.

Example (3): Another example is 2326 – 1875. Here 1875 becomes 2000 by adding 125 and 2326 becomes 2451 by adding 125. The number becomes 2451 – 2000= 451. Here the subtraction 2451 – 2000 is easier than the subtraction 2326 – 1875. The answer of both is same 451.

Example (4): The subtraction of 3786 – 2998. Here 2998 becomes 3000 by adding 2 and 3786 becomes 3788 by adding 2. The problem of 3788 – 3000 is easier than 3786 – 2998 and our answer is 788. This answer is same for both the problems.

Fast Math and Shortcut Rules for Subtraction- Rule-II:

Useful Shortcut rules for Subtraction which was more helpful in the fast math to solve the aptitude questions were given below. Candidates those who are preparing for the banking and all competitive exams can use this.

Rule: II. Double Column Addition and Subtraction Method:

This following method is works when there is a series of additions and subtractions are to be performed in a line.

Example (1):

1026

–  4572

+ 5263

–  2763

+ 8294

_____________

Explanation: We have to look the signs given before the numbers and then start adding and adding and subtracting from the top right position.

Step I: First Double Column

26 – 72= – 46, – 46 + 63= 17,

17 – 63= – 46, – 46 + 94= 48.

Step II: Second Double Column

10 – 45= – 35, – 35 + 52= 17,

17 – 27= – 10, – 10 + 82= 72.

Now Combine the Step II and Step I.

You will get the same answer if you also use the normal method.

Example (2):

7676

–  1431

+ 5276

–  3489

+ 1546

_____________

Explanation: We have to look the signs given before the numbers and then start adding and adding and subtracting from the top right position.

Step I: First Double Column

76 – 31= 45, 45 + 76= 121 (here in the 121 take the last two digits from 121 i.e., 21)

21 – 89= – 68, – 68 + 46= -22 (Here the answer comes in minus so add 100 with the answer)

100 + (- 22) = 78.

Step II: Second Double Column

76 – 14= 62, 62 + 52= 114 (Take the last two digits from 114 i.e., 14)

14 – 34= – 20, – 20 + 15= – 5 (Here the answer comes in minus so add 100 with the answer)

100 + (-5) = 95

Now Combine the Step II and Step I.

EXAMPLE 1: Solve 12 + 22 ÷ 11 × (18 ÷ 3)^2 – 10
= 12 + 22 ÷ 11 × 6^2 – 10 (Brackets first)
= 12 + 22 ÷ 11 × 36 – 10 (Exponents)
= 12 + 2 × 36 – 10 = 12 + 72 – 10 (Division and multiplication, left to right)
= 84 – 10 = 74 (Addition and Subtraction, left to right)
EXAMPLE 2: Solve 4 + 10 – 3 × 6 / 3 + 4
= 4 + 10 – 18/3 + 4 = 4 + 10 – 6 + 4 (Division and multiplication, left to right)
= 14 – 6 + 4 = 8 + 4 = 12 (Addition and Subtraction, left to right)

## To Solve Modulus of a Real Number

The Modulus (or the absolute value) of x is always either positive or zero, but never negative. For any real number x, the absolute value or modulus of x is denoted by |x| and is defined as

|x|x {if ≥ 0} and x {if 0}
EXAMPLE 1: Solve |8|
|8| = |-8|  = 8

### Tips to Crack Approximation

Conversion of decimal numbers to nearest number
To solve such questions, first convert the decimal to nearest value. Then simplify the given equation using the new values that you have obtained.
EXAMPLE 1: Solve 4433.764 – 2211.993 – 1133.667 + 3377.442
Here,
4433.764 = 4434
2211.993 = 2212
1133.667 = 1134
3377.442 = 3377
Now simplify, 4434 –  2212 – 1134 + 3377 = 4466

EXAMPLE 2: Solve 530 x 20.3% + 225 x 16.8%
Here, 20.3% becomes 20% and 16.8% becomes 17%
Now, simplify 530 x 20% + 225 x 17%
= 106 + 38.25 = 144.25
Approximation of Square Roots
1. To simplify a square root, you can follow these steps:
2. Factor the number inside the square root sign.
3. If a factor appears twice, cross out both and write the factor one time to the left of the square root sign. If the factor appears three times, cross out two of the factors and write the factor outside the sign, and leave the third factor inside the sign. Note: If a factor appears 4, 6, 8, etc. times, this counts as 2, 3, and 4 pairs, respectively.
4. Multiply the numbers outside the sign.
5. Multiply the numbers left inside the sign.
6. To simplify the square root of a fraction, simplify the numerator and simplify the denominator.

Now we are going to share some important tips and tricks that will help you prepare the Simplification – Approximation topic better.

## Simplification / Approximation: Tips and Tricks

We strictly recommend you to learn square (up to 30) and cube (up to 20).We will discuss here methods to solve and types of problems which are generally asked in exams.

## Unit Digits and its applications

Ex: 298: 8 is the unit place in 298.
Ex: 1947: 7 is the unit place in 1847.

Ex: 2345×6789
(A)15920206 (B)15920208 (C) 15920205 (D) 15920204
Solution: When unit place of 5 in 2345 and unit place of 9 in 6789 multiplies we will get 45. So when both numbers are multiplies it should have 5 at its unit place which is only in option C.
Ex: 43 × 36 + 57 × 89

(A)6380 (B)5728 (C)6782 (D)6621
The unit digit will be the sum of the individual unit digits.
(3×6)+(7×9) = 18+63 = 81
So the resultant number must have 1 at its unit place.

### Digit Sum

It is the sum of all digits of the number used in making the number and keep adding till we have only one digit left.
Ex: 2345
Digit sum = (2+3+4+5) = 14 = 1+4 = 5

Ex: 123456789

Digit sum = (1+2+3+4+5+6+7+8+9) = 45 = (4+5) = 9
Note: In this case our assumption is that 9 should be treated as 0.

Ex: 123 × 456 × 781
(A)43804728 (B) 53804728 (C) 53804528 (D)33804958
LHS (Digit sum)= (1+2+3)×(4+5+6)×(7+8+1)= 6× 6× 7 = 36× 7 = 9 × 7 = 63 = 0
RHS (Digit sum):
(A) (4+3+8+0+4+7+2+8)= 36 =(3+6)= 9 = 0
(B) (5+3+8+0+4+7+2+8) = 37 = 10 = (1+0) = 1
(C) = 35 = (3+5) = 8
(D) =31 = (3+1) = 4
So, Option A is the answer.
Ex: 2011×97+50123 = ? × 743
(A) 340 (B) 330 (C) 350 (D) 303 (E) 345
Solution:
In LHS 2011×97, unit digit will be 7
In 50123, the unit digit is 3, So when we add these, the addition will have ‘0’ at its unit place.
In RHS, we also need ‘0’ at the unit place, the number which has to multiplied by 743 must consist 0 at its unit place. So, option (D) and (E) are eliminated.
Now Let’s apply Unit digit and digit sum
In LHS, 2011×97+ 50123
4 × 7 + 11 = 28+11 = 10+2 = 1+2 = 3
In RHS if option is (A)
then 340 × 743 = 7×14 = 7 × 5 = 35 = 8
LHS ≠ RHS
In RHS if option is (B)
then 330 × 743 = 6× 14 = 6× 5 = 30 = 3
LHS = RHS, It is the answer. If you check other options it will not satisfy this.
Ex: 6269+0.75× 4444+0.8×185 =?
(A)9759 (B)9750 (C)9740 (D)9755 (E)9655
Solution:
6269+ (3/4)×4444+148.0
6269+3333+148
We can see that unit digit is Zero. So options remained are B and C.
Now, (23)+(12)+(13)
5+3+4 = 12 = 3
Applying digit sum for (C) = 2 and (B) = 3

### Perfect Square

 If the square ends in 1 4 5 6 9 0 The number would end in 1,9 2,8 5 4,6 3,7 0
When a number is given, split it in two parts, in such a way that 2nd part has last two digits of number and first part will have remaining digits.
Ex 1:  Square root of 3481
Split number in two parts i.e. 34 and 81(last two digits)
We know that square of number ends in 1, so square root ends either in 1 or 9.
Check, 34 lies between 25 (square of 5) and 36 (square of 36). Take smaller number.
So, our answer is either 51 or 59.
but we know 502 = 2500 and 602 = 3600, 3481 is nearest to 3600. So the answer is 59.
or 34 is more close to 36 than 25, so the answer is 59.
Ex 2: 76176
Split: 761  76
Number will end in either 4 or 6,
729(272) < 761 < 784 (282), So the answer may be 274  or 276. 761 is more close to 784, so the answer is 276.
Ex 3: square root of 75076
Split: 750  76
Number will end in either 4 or 6
729(272) < 750 < 784 (282), So the answer may be 274  or 276. 750 is more close to 729 than 784, so the answer is 274.
Non-Perfect Square: This gives approximate value not an exact value.
Ex4: 1000

961(312) < 1000 < 1024(322)
Now, 1000 is nearest to 1024
So, 32 – ((1024-1000)/(2× 32))
32 – (24/64)
32-.375 = 31.625
or 31+((1000-961)/(2× 31))
31 + (39/62)
31+.629 ≈ 31.63

## How to calculate Cube root?

 If the cube ends in 1 2 3 4 5 6 7 8 9 0 The number would end in 1 8 7 4 5 6 3 2 9 0
When a number is given, split it in two parts, in such a way that 2nd part has last three digits of number and first part will have remaining digits.

Ex 1: cube root of 3112136
Split in two parts  3112    136
Number will end with 6
143 (2744) < 3112 < 153 (3375)
Choose the smaller number and answer will be 146.
Ex 2: cube root of 2406104
split in two parts 2406   104
Number will end with 4
133 (2197) < 2406 < 143(2744)
So the answer will be 134.

While dealing with calculation intensive sections like Quant and DI, it is very important to pick the right questions use the best methods. As we grow, we tend to get habituated in our day to day calculations and employ conventional methods without thinking whether it is the best for a given scenario. Many a time there will be an approach that is much easier than the conventional methods to solve a given problem. Most of the so called calculation intensive questions are not that scary if we think a bit before solving them. As a rule of thumb, always spend few seconds to identify the best approach before start solving.

Some useful methods are given below which can help in our calculations.

To approximate Actual values

If (and only if) we need to find the actual value of a given fraction, represent the numerator as sum or difference of terms related to denominator.

1449/132 =
1449 = 1320 + 132 – 3
1449/132 = 10 + 1 – a small value ≈ little less than 11 (actual value is 10.977)

36587 / 123 =
36587 = 36900 – 246 – 61.5 – …
36587 / 123 = 300 – 2 – 0.5 – a small value ≈ little less than 297.5 (actual is 297.455)

1569 / 12 =
1569 = 1200 + 360 + 8.4 + 0.6
1569 / 12 = 100 + 30 + 0.7 + 0.05 = 130.75

This method should suffice for the level of accuracy expected in our exams.

Another method is to reduce the complexity of fraction and then solve. Complexity of a fraction can be directly related to the complexity of its denominator. If we simplify denominator, we simplify the fraction. Add to or subtract from the denominator to make it an easier value (like add 2 to 1998 to get 2000 or subtract 16 from 116 to get 100).

While adjusting the denominator always remember to BALANCE the fraction. Balancing fraction is not just adding/subtracting the same number to/from the numerator that we used to change the denominator.

Consider a fraction p/q = n; then p = qn.

If we add a number x to q, we need to add nx to p to balance the fraction. Also if q is reduced by a number x, p needs to be reduced by nx.

Here the approximation comes while fixing n. If the given options are separated well enough from each other and simplification of denominator is pretty obvious, then this method can be employed. If we have closer options it is better to stick with the method we discussed first.

1569 / 12 = ?

Here if we make the denominator as 10 we can tell the value in no time. To do so, we need to subtract 2 from denominator. Numerator is more than 130 times the denominator (n ≈ 130).  Hence to balance the fraction we need to subtract 2 * 130 from numerator.

1569 / 12 ≈ 1309 / 10 ≈ 130.9 (actual value is 130.75)

To Approximate relative values

Most of the DI questions revolves around sorting the given numbers/fractions or finding its relative position (lesser/greater than) based on a reference value. If we don’t need the actual value, DON’T find the actual value.

Find the largest and smallest value among the below fractions
56/298, 46/374, 138/493, 37/540, 670/2498

We will do the first level approximation by guesstimating the given fractions. Try to represent the given numbers in 1/x format. While arranging fractions we usually try to represent the given fractions with the same denominator after finding the LCM of all denominators. But we are here to solve faster using approximation. We will take an easier route, Make the numerator same, i.e. one.

56/298, we know 56 * 6 > 298 = > 56/298 > 1/6. Note that we didn’t find the actual value of 56 * 6; we just want to get the closest multiple of 56 to the number 298.

56/298 = Greater than 1/6
46/374= Less than 1/8
138/493 = Greater than 1/4
37/540 = Greater than 1/15
670/2498 = Greater than 1/4

We don’t have any confusion in finding the smallest which is 37/540 (1/15 is less than other values). But we have 2 candidates fighting for the largest fraction title, 138/493 and 670/2498. We will consider only those two and try to get an approximate value. We will try both methods discussed before for finding the actual value.

Method 1:

138 = 98.6 + 24.65 + 12.325 + …
138/493 ≈ 0.2 + 0.05 + 0.025 + small value ≈ greater than 0.275

670 = 499.6 + 124.9 + 49.96 – 4.46
670/2498 ≈ 0.2 + 0.5 + 0.02 – small value ≈ less than 0.27

Hence 138/493 is the largest.

Method 2:

138/493,

We can see denominator is close to 3.5 times numerator. Hence if we increase denominator by x, we need to balance the fraction by increasing numerator by x/3.5. We will get an easier fraction if we can write denominator as 500 by adding 7. We also need to add 7/3.5 = 2 to the numerator.

138/493 ≈ 140/500 ≈ 0.28

Similarly for 670/2498, here we can get a neat fraction by adding 2 to the denominator. And here as 2 is negligible compared to the denominator we can very well skip the balancing part and write fraction as 670/2500 = 0.268

Hence, 138/493 is the largest.

Here we wrote 670/2500 = 0.268. How?

670/2500 = 67/250, we can get denominator as 1000 by multiplying both sides by 4. Hence 67/250 = 268/1000 = 0.268

We used the same logic while ‘cleaning up’ 140/500. Multiply both sides with 2 to get denominator as 1000. Fraction becomes 280/1000 = 0.028

Here, instead of finding actual values of all five fractions and comparing them we just played with the relative values of the fractions and found actual values only for two cases which were required to get the answer.

Another usual DI question type is to find the relative position of a given value based on a reference value. This question comes like ‘How many students scored marks more than class average (Reference value)’ , ‘How many players has strike rate higher than Sachin (Reference value)’ etc…

How many of the given values are greater than 0.7

11/13, 25/34, 33/46, 44/65, 56/81

As we are asked to find only the relative values (with respect to 0.7) don’t jump into finding actual values. Take few seconds to write the below statement which will help us in solving faster.

If x/y > 0.7, x > 0.7 y, 10x > 7y

So we need to find all fractions where 10 times numerator is greater than 7 times y. multiplying both sides with 10 is to ease the calculation and simplify the comparison

Take fractions one by one

Three fractions (11/13, 25/34 and 33/46) are greater than 0.7

Most of us have higher comfortable level with multiplication than division. To find relative values based on a reference point, convert division into multiplication. This way we can get our answers faster without messing with our accuracy.

In our example 56/81 = 0.69, still we were able to find it is lesser than 0.7 without doing any complicated or time consuming stuff.  Sweet, right!

EXAMPLE 1: Solve 12 + 22 ÷ 11 × (18 ÷ 3)^2 – 10

= 12 + 22 ÷ 11 × 6^2 – 10 (Brackets first)
= 12 + 22 ÷ 11 × 36 – 10 (Exponents)
= 12 + 2 × 36 – 10 = 12 + 72 – 10 (Division and multiplication, left to right)
= 84 – 10 = 74 (Addition and Subtraction, left to right)

EXAMPLE 2: Solve 4 + 10 – 3 × 6 / 3 + 4

= 4 + 10 – 18/3 + 4 = 4 + 10 – 6 + 4 (Division and multiplication, left to right)
= 14 – 6 + 4 = 8 + 4 = 12 (Addition and Subtraction, left to right)

To Solve Modulus of a Real Number

The Modulus (or the absolute value) of x is always either positive or zero, but never negative. For any real number x, the absolute value or modulus of x is denoted by |x| and is defined as
|x|x {if ≥ 0} and x {if 0}
EXAMPLE 1: Solve |8|
|8| = |-8|  = 8

Conversion of decimal numbers to nearest number

To solve such questions, first convert the decimal to nearest value. Then simplify the given equation using the new values that you have obtained.

EXAMPLE 1: Solve 4433.764 – 2211.993 – 1133.667 + 3377.442

Here,
4433.764 = 4434
2211.993 = 2212
1133.667 = 1134
3377.442 = 3377

Now simplify, 4434 –  2212 – 1134 + 3377 = 4466

EXAMPLE 2: Solve 530 x 20.3% + 225 x 16.8%

Here, 20.3% becomes 20% and 16.8% becomes 17%
Now, simplify 530 x 20% + 225 x 17%
= 106 + 38.25 = 144.25

Approximation of Square Roots

(1) To simplify a square root, you can follow these steps:

(2) Factor the number inside the square root sign.

(3) If a factor appears twice, cross out both and write the factor one time to the left of the square root sign. If the factor appears three times, cross out two of the factors and write the factor outside the sign, and leave the third factor inside the sign. Note: If a factor appears 4, 6, 8, etc. times, this counts as 2, 3, and 4 pairs, respectively.

(4) Multiply the numbers outside the sign.

(5) Multiply the numbers left inside the sign.

(6) To simplify the square root of a fraction, simplify the numerator and simplify the denominator.

NOTE: Check that the outside number squared times the inside number should equal the original number inside the square root.

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