12/5/24, 11:57 AM Fast Multiplication Techniques

CATEGORY 1: Multiplying by Specific Numbers

1. Multiplying by 5

Shortcut: Divide by 2, then multiply by 10.

Example: 68 × 5 = 68 ÷ 2 = 3468×5=68÷2=34, then 34 × 10 = 340
34×10=340.

2. Multiplying by 9

Shortcut: Multiply by 10, then subtract the original number.

Example: 43 × 9 = 43 × 10 − 43 = 430 − 43 = 387
43×9=43×10−43=430−43=387.

3. Multiplying by 11

Shortcut for Two-Digit Numbers:

1. Add the digits of the number.
2. Place the sum between the digits.
Example: 54 × 11 = 5 ( 5 + 4 ) 4 = 59454×11=5(5+4)4=594.
For larger numbers, apply digit-by-digit shifting.

4. Multiplying by Powers of 2

Double the number for each power of 2:

Example: 23 × 16 = ( ( ( 23 × 2 ) × 2 ) × 2 ) × 2 = 36823×16=
(((23×2)×2)×2)×2=368.

5. Multiplying by Numbers Ending in Zero

Ignore the zeros, multiply the rest, then add the zeros back.
Example: 300 × 40 = ( 3 × 4 ) 300×40=(3×4) and append three zeros:
12 , 00012,000.

CATEGORY 2: Numbers Near Base (10, 100, etc.)

6. Numbers Close to 100

2 2
Use ( 𝑎 + 𝑏 ) ( 𝑎 − 𝑏 ) = 𝑎 − 𝑏 (a+b)(a−b)=a2−b2:
Example:
2 2
97 × 103 = ( 100 − 3 ) ( 100 + 3 ) = 100 − 3 = 10 , 000 − 9 = 9 , 991
97×103=(100−3)(100+3)=1002−32=10,000−9=9,991.

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7. Numbers Near Powers of 10

2
For ( 𝑏𝑎𝑠𝑒 + 𝑥 ) ( 𝑏𝑎𝑠𝑒 + 𝑦 ) = 𝑏𝑎𝑠𝑒 + 𝑏𝑎𝑠𝑒 ( 𝑥 + 𝑦 ) + 𝑥𝑦(base+x)
(base+y)=base2+base(x+y)+xy:
2
Example: 108 × 112 = 100 + 100 ( 8 + 12 ) + ( 8 × 12 ) = 12 , 096
108×112=1002+100(8+12)+(8×12)=12,096.

CATEGORY 3: Vedic Math Techniques

8. Vertically and Crosswise Method

Multiply diagonally and vertically for two numbers:

Example:
23 × 45 = ( 2 × 4 ) ∥ ( 2 × 5 + 3 × 4 ) ∥ ( 3 × 5 ) = 900 + 350 + 15 = 1035
23×45=(2×4)∥(2×5+3×4)∥(3×5)=900+350+15=1035.

9. Multiplication Using Base and Deviations

Use deviations from a base:

Example:
104 × 97 = 100 + ( 4 + ( − 3 ) ) + ( 4 × − 3 ) = 10000 − 12 = 9 , 988
104×97=100+(4+(−3))+(4×−3)=10000−12=9,988.

CATEGORY 4: Advanced Patterns

10. Squaring Numbers Ending in 5

2
Formula: 𝑛 = ( 𝑎 × ( 𝑎 + 1 ) ) + 25n2=(a×(a+1))+25, where 𝑎a is the first
part of the number.
2
Example: 35 = 3 × 4 = 12 , append 25 : 1225
352=3×4=12, append 25:1225.

11. Squaring Any Two-Digit Number

2 2 2
Use ( 𝑎 + 𝑏 ) = 𝑎 + 2𝑎𝑏 + 𝑏 (a+b)2=a2+2ab+b2:
2 2 2
Example: 43 = 40 + 2 ( 40 ) ( 3 ) + 3 = 1600 + 240 + 9 = 1849
432=402+2(40)(3)+32=1600+240+9=1849.

12. Multiplying Two Numbers Just Below or Above 50

Formula: ( 50 − 𝑎 ) ( 50 − 𝑏 ) = 2500 − 50 ( 𝑎 + 𝑏 ) + 𝑎𝑏(50−a)

(50−b)=2500−50(a+b)+ab:
Example: 46 × 48 = 2500 − 50 ( 6 + 8 ) + ( 6 × 8 ) = 2208
46×48=2500−50(6+8)+(6×8)=2208.

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CATEGORY 5: Distributive Techniques

13. Breaking Down Numbers for Simplicity

Example: 64 × 57 = ( 64 × 50 ) + ( 64 × 7 ) = 3200 + 448 = 364864×57=

(64×50)+(64×7)=3200+448=3648.

14. Chunking Large Numbers

Example: 234 × 567 = ( 234 × 500 ) + ( 234 × 60 ) + ( 234 × 7 )

234×567=(234×500)+(234×60)+(234×7).

CATEGORY 6: Digit Manipulation

15. Doubling and Halving

Double one number and halve the other for easier calculation:
Example: 32 × 25 = ( 16 × 50 ) = 80032×25=(16×50)=800.

16. Multiplication of Fractions

Simplify and multiply directly.

Example: 3/5 × 25 = 153/5×25=15.

CATEGORY 7: Special Multiplications

17. Multiplying by 99, 999, etc.

Subtract the number from 100 , 1000 , etc.100,1000,etc.:

Example: 87 × 99 = ( 87 × 100 ) − 87 = 8700 − 87 = 861387×99=
(87×100)−87=8700−87=8613.

18. Multiplying Numbers with Recurring Decimals

Example: 0.333 × 9 = ( 1/3 × 9 ) = 30.333×9=(1/3×9)=3.

19. Multiplying by 15

Shortcut: Multiply by 10 and add half the number.

Example: 46 × 15 = 460 + 230 = 69046×15=460+230=690.

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CATEGORY 8: Multiplication Tricks for Specific Cases

20. Multiplication by 1.25, 1.5, etc.

Convert to fractions:
Example: 64 × 1.25 = 64 × 5/4 = 8064×1.25=64×5/4=80.

21. Multiplying by Two Numbers with a Common Base

2 2
Use 𝑎 − 𝑏 a2−b2:
2 2
Example: ( 12 + 2 ) ( 12 − 2 ) = 12 − 2 = 144 − 4 = 140(12+2)
(12−2)=122−22=144−4=140.

22. Multiplying Mixed Numbers

Convert to improper fractions, then multiply.

1 2 7 22 154
Example: 2 3 × 4 5 = 3
× 5
= 15
= 10.27231​×452​=37​×522​=15154​
=10.27.

CATEGORY 9: Techniques for Large Numbers

23. Fast Approximation

Round numbers, multiply, and adjust:

Example: 498 × 102 = ( 500 × 102 ) − ( 2 × 102 ) = 50 , 796
498×102=(500×102)−(2×102)=50,796.

24. Multiplication by Complementary Numbers

For ( 𝑎 ) ( 𝑏 + 𝑐 ) = ( 𝑎 × 𝑏 ) + ( 𝑎 × 𝑐 ) (a)(b+c)=(a×b)+(a×c):
Example: 54 × 106 = ( 54 × 100 ) + ( 54 × 6 ) 54×106=(54×100)+
(54×6).

CATEGORY 10: Other Advanced Techniques

25. Cross Multiplication

Multiply diagonally and sum:

Example: 23 × 4523×45:
1. Multiply 2 × 5 = 102×5=10, 4 × 3 = 124×3=12.
2. Sum 1000 + 20 = 10201000+20=1020.

With practice, these techniques will drastically cut down your calculation time during
exams. Let me know if you’d like examples or drills!

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12/5/24, 3:17 PM Fast Multiplication Techniques

give me rules for division shortcut

Here’s a comprehensive list of division shortcuts and rules for competitive exams.
These rules simplify division calculations, enabling faster problem-solving:

CATEGORY 1: Division by Specific Numbers

1. Division by 5

Shortcut: Multiply the number by 2, then divide by 10.

Example: 245 ÷ 5 = ( 245 × 2 ) ÷ 10 = 490 ÷ 10 = 49245÷5=
(245×2)÷10=490÷10=49.

2. Division by 9

Shortcut: Use the sum of the digits to check divisibility and find remainders.
Example: 728 ÷ 9728÷9: Add digits 7 + 2 + 8 = 177+2+8=17, repeat
1 + 7 = 81+7=8. Quotient: 8080, remainder: 88.

3. Division by 11

Shortcut: Alternate the sum of digits to check divisibility.

Example: 121 ÷ 11121÷11: 1 − 2 + 1 = 01−2+1=0, so perfectly divisible,
quotient is 1111.

CATEGORY 2: Division by Powers of 2

4. Division by 2, 4, 8, etc.

Shortcut: Halve the number repeatedly for each power of 2.

Example: 64 ÷ 8 = ( ( 64 ÷ 2 ) ÷ 2 ) ÷ 2 = 864÷8=((64÷2)÷2)÷2=8.

CATEGORY 3: Division by Numbers Ending in Zero

5. Ignore Trailing Zeros

Shortcut: Remove the zeros temporarily, divide, then adjust.

Example: 3000 ÷ 40 = ( 300 ÷ 4 ) = 753000÷40=(300÷4)=75.

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CATEGORY 4: Vedic Math Techniques

6. Divisibility Rules

Shortcut: Check for divisibility before performing full division:

3: Sum of digits divisible by 3.
4: Last two digits divisible by 4.
6: Divisible by both 2 and 3.
8: Last three digits divisible by 8.
9: Sum of digits divisible by 9.
11: Difference of sums of alternate digits divisible by 11.

7. Nikhilam Method

Shortcut: Use deviations from a base like 10, 100, etc., for faster calculations.
Example: 997 ÷ 999997÷999: − 2/999 = 1−2/999=1 approximately.

CATEGORY 5: Advanced Patterns

8. Division by Numbers Near 100

Shortcut: Approximate by shifting the decimal and adjusting.

Example: 125 ÷ 99 ≈ 1.26125÷99≈1.26.

9. Long Division Shortcut

Shortcut: Divide sequentially, carrying remainders forward.

Example: 987 ÷ 32987÷32: 32 × 30 = 96032×30=960, remainder 2727.

CATEGORY 6: Fraction Conversion Techniques

10. Simplify Fractions

Shortcut: Reduce fractions to their simplest form by dividing numerator and

denominator by their greatest common divisor (GCD).
Example: 45 ÷ 60 = 3/445÷60=3/4.

11. Recurring Decimals

Shortcut: Identify recurring patterns for repeating decimals.

Example: 1/3 = 0.333...1/3=0.333...; 2/3 = 0.666...2/3=0.666....

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CATEGORY 7: Approximations

12. Approximation for Large Numbers

Shortcut: Round the divisor, divide, then adjust for accuracy.

Example: 98765 ÷ 98 ≈ 98765 ÷ 100 = 987.65
98765÷98≈98765÷100=987.65.

CATEGORY 8: Special Cases

13. Division by Multiples of 10

Shortcut: Remove the zero(s), divide, then adjust.

Example: 4500 ÷ 50 = ( 4500 ÷ 10 ) ÷ 5 = 450 ÷ 5 = 904500÷50=
(4500÷10)÷5=450÷5=90.

14. Division by Decimal Numbers

Shortcut: Multiply numerator and denominator by 10, 100, etc., to remove

decimals.
Example: 4.5 ÷ 0.9 = ( 45 ÷ 9 ) = 54.5÷0.9=(45÷9)=5.

CATEGORY 9: Combining Multiplication and Division

15. Simplify Using Proportionality

Shortcut: Simplify ratios before dividing.

Example: 120 ÷ 15 = ( 120 ÷ 5 ) ÷ 3 = 24 ÷ 3 = 8120÷15=
(120÷5)÷3=24÷3=8.

16. Doubling and Halving

Shortcut: Double one number and halve the other for easier calculations.
Example: 64 ÷ 16 = ( 32 ÷ 8 ) = 464÷16=(32÷8)=4.

Master these rules with practice, and your division calculations will become
extremely efficient! Would you like me to create a printable version of these
shortcuts?

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12/5/24, 4:01 PM Fast Multiplication Techniques

Here are the fastest addition shortcuts for competitive exams, categorized for ease of
use:

CATEGORY 1: Breaking Numbers for Easy Addition

1. Split and Combine

Shortcut: Break numbers into simpler components.

Example: 247 + 368 = ( 240 + 360 ) + ( 7 + 8 ) = 600 + 15 = 615
247+368=(240+360)+(7+8)=600+15=615.

2. Grouping Similar Digits

Shortcut: Add digits of the same place value first.

Example: 123 + 456 = ( 100 + 400 ) + ( 20 + 50 ) + ( 3 + 6 ) = 579
123+456=(100+400)+(20+50)+(3+6)=579.

3. Left-to-Right Addition

Shortcut: Add higher place values first for mental clarity.

Example:
275 + 438 = ( 200 + 400 ) + ( 70 + 30 ) + ( 5 + 8 ) = 600 + 100 + 13 = 713
275+438=(200+400)+(70+30)+(5+8)=600+100+13=713.

CATEGORY 2: Adding Close Numbers

4. Near Base Numbers

Shortcut: Use deviations from a common base.

Example:
98 + 103 = 100 + ( 98 − 100 ) + ( 103 − 100 ) = 100 − 2 + 3 = 201
98+103=100+(98−100)+(103−100)=100−2+3=201.

5. Numbers Close to Multiples of 10

Shortcut: Round and adjust.

Example: 89 + 47 = ( 90 + 50 ) − 1 − 3 = 13789+47=(90+50)−1−3=137.

6. Adding Numbers Close to 100

Shortcut: Add the base and deviations.

Example: 96 + 104 = ( 100 − 4 ) + ( 100 + 4 ) = 20096+104=(100−4)+
(100+4)=200.

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CATEGORY 3: Column Addition for Multiple Numbers

7. Vertical Summation

Shortcut: Add column-wise for multi-digit numbers.

Example:

diff

245
+ 367
------
612

8. Pairing Technique

Shortcut: Pair numbers to form round totals first.

Example:
25 + 47 + 35 + 53 = ( 25 + 35 ) + ( 47 + 53 ) = 60 + 100 = 160
25+47+35+53=(25+35)+(47+53)=60+100=160.

9. Adding Sequences

Shortcut: Use arithmetic progression formulas if numbers form a sequence.

Example: Sum of 1 + 2 + 3 + ... + 10 = 𝑛 ( 𝑛 + 1 ) /2 = 10 ( 11 ) /2 = 55
1+2+3+...+10=n(n+1)/2=10(11)/2=55.

CATEGORY 4: Advanced Patterns

10. Adding Fractions

Shortcut: Convert fractions to a common denominator before adding.

Example: 1/4 + 3/8 = 2/8 + 3/8 = 5/81/4+3/8=2/8+3/8=5/8.

11. Complementary Addition

Shortcut: Add complements to simplify.

Example: 63 + 29 = ( 63 + 30 ) − 1 = 9263+29=(63+30)−1=92.

12. Digit by Digit Addition

Shortcut: Add digits systematically, adjusting for carryovers.

Example: 57 + 68 = 5 + 6 ( 11 ) , 7 + 8 ( 15 , 𝑐𝑎𝑟𝑟𝑦1 ) = 125
57+68=5+6(11),7+8(15,carry1)=125.

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CATEGORY 5: Special Techniques

13. Repeated Numbers

Shortcut: Multiply for identical repeated additions.

Example: 75 + 75 + 75 = 3 × 75 = 22575+75+75=3×75=225.

14. Adding Recurring Decimals

Shortcut: Convert to fractions before adding.

Example: 0.333 + 0.666 = ( 1/3 ) + ( 2/3 ) = 10.333+0.666=(1/3)+
(2/3)=1.

15. Adding Numbers with Decimals

Shortcut: Align decimal points, then add column-wise.

Example:

diff

45.67
+ 23.89
-------
69.56

CATEGORY 6: Mental Math for Large Numbers

16. Approximation for Large Numbers

Shortcut: Round numbers, add, then adjust.

Example:
9876 + 5432 = ( 9900 + 5400 ) − 24 = 15 , 300 − 24 = 15 , 276
9876+5432=(9900+5400)−24=15,300−24=15,276.

17. Chunking

Shortcut: Add numbers in chunks for simplicity.

Example:
1452 + 6789 = ( 1450 + 6780 ) + ( 2 + 9 ) = 8239 + 11 = 8250
1452+6789=(1450+6780)+(2+9)=8239+11=8250.

18. Adding Overlapping Groups

Shortcut: Find common parts and adjust.

Example: 123 + 234 = ( 100 + 200 ) + ( 20 + 30 ) + ( 3 + 4 ) = 357
123+234=(100+200)+(20+30)+(3+4)=357.

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CATEGORY 7: Vedic Math Techniques

19. Ekadhikena Purvena (One More Than the Previous)

Shortcut: Use for series-based additions.

Example: Sum of 1 + 2 + 3 + ... + 𝑛 = 𝑛 ( 𝑛 + 1 ) /21+2+3+...+n=n(n+1)/2.

20. Digital Sum Check

Shortcut: Quickly verify the sum by adding individual digits of all numbers.
Example: 236 + 419 = 655236+419=655 (digit sum:
2 + 3 + 6 = 11 ; 4 + 1 + 9 = 14 ; 6 + 5 + 5 = 16
2+3+6=11;4+1+9=14;6+5+5=16).

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For substraction

Here are the fastest subtraction shortcuts for competitive exams to enhance
calculation speed:

CATEGORY 1: Subtracting Specific Numbers

1. Subtracting from Multiples of 10

Shortcut: Borrow from the nearest multiple of 10.

Example: 60 − 47 = ( 60 − 50 ) + 3 = 1360−47=(60−50)+3=13.

2. Subtracting by Adding

Shortcut: Convert subtraction into addition using complements.

Example: 100 − 72 = 28100−72=28, as 72 + 28 = 10072+28=100.

3. Subtracting Numbers Near Multiples of 10

Shortcut: Adjust the numbers to simplify.

Example: 98 − 43 = ( 100 − 43 ) − 2 = 5598−43=(100−43)−2=55.

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CATEGORY 2: Breaking Numbers for Simplicity

4. Left-to-Right Subtraction

Shortcut: Subtract higher place values first for clarity.

Example: 734 − 268 = ( 700 − 200 ) + ( 30 − 60 ) + ( 4 − 8 ) = 466
734−268=(700−200)+(30−60)+(4−8)=466.

5. Chunking

Shortcut: Break one number into manageable parts.

Example: 645 − 389 = ( 645 − 300 ) − 89 = 345 − 89 = 256645−389=
(645−300)−89=345−89=256.

6. Balancing

Shortcut: Balance both numbers to simplify subtraction.

Example: 1007 − 598 = ( 1007 + 2 ) − ( 598 + 2 ) = 1009 − 600 = 409
1007−598=(1007+2)−(598+2)=1009−600=409.

CATEGORY 3: Borrowing and Adjusting

7. Complement Subtraction

Shortcut: Use the complement method for easier calculations.

Example: 900 − 387 = ( 900 − 400 ) + 13 = 513900−387=
(900−400)+13=513.

8. Subtracting Large Numbers

Shortcut: Split into smaller steps.

Example: 7854 − 2347 = ( 7854 − 2000 ) − 347 = 5854 − 347 = 5507
7854−2347=(7854−2000)−347=5854−347=5507.

CATEGORY 4: Numbers Close to Base Values

9. Numbers Close to 100

Shortcut: Use deviations from 100.

Example: 105 − 97 = ( 105 − 100 ) − ( 97 − 100 ) = 8105−97=
(105−100)−(97−100)=8.

10. Numbers Close to 1000

Shortcut: Adjust numbers by rounding.

Example: 995 − 878 = ( 995 − 900 ) + ( 900 − 878 ) = 95 + 22 = 117
995−878=(995−900)+(900−878)=95+22=117.

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CATEGORY 5: Vedic Math Techniques

11. All from 9 and Last from 10

Shortcut: For numbers with zeros at the end, subtract each digit from 9 except
the last one, which is subtracted from 10.
Example: 1000 − 357 = ( 9 − 3 ) ( 9 − 5 ) ( 10 − 7 ) = 6431000−357=
(9−3)(9−5)(10−7)=643.

12. Ekadhikena Purvena (One More Than the Previous)

Shortcut: Use for series-based subtractions.

Example: 10000 − 6758 = ( 10000 − 7000 ) + ( 7000 − 6758 ) = 3242
10000−6758=(10000−7000)+(7000−6758)=3242.

CATEGORY 6: Decimal Subtraction

13. Align Decimal Points

Shortcut: Line up decimals and subtract column-wise.

Example: 45.67 − 23.89 = ( 45 − 23 ) + ( 0.67 − 0.89 ) = 21.78
45.67−23.89=(45−23)+(0.67−0.89)=21.78.

14. Subtracting Recurring Decimals

Shortcut: Convert recurring decimals to fractions.

Example: 0.666 − 0.333 = ( 2/3 ) − ( 1/3 ) = 1/3 = 0.3330.666−0.333=
(2/3)−(1/3)=1/3=0.333.

CATEGORY 7: Mental Math for Subtraction

15. Approximation

Shortcut: Round numbers, subtract, and adjust.

Example: 8765 − 5432 = ( 8800 − 5400 ) − 35 = 33658765−5432=
(8800−5400)−35=3365.

16. Digit-by-Digit Subtraction

Shortcut: Subtract digits column-wise, handling carryovers.

Example: 648 − 279 = ( 6 − 2 ) ( 4 − 7 + 10 ) ( 8 − 9 + 10 ) = 369
648−279=(6−2)(4−7+10)(8−9+10)=369.

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CATEGORY 8: Special Cases

17. Subtracting Fractions

Shortcut: Convert to a common denominator.

Example: 5/8 − 3/8 = ( 5 − 3 ) /8 = 2/8 = 1/45/8−3/8=(5−3)/8=2/8=1/4.

18. Subtracting Repeated Numbers

Shortcut: Multiply subtractions if numbers repeat.

Example: 75 − 75 − 75 = 3 × ( − 75 ) = − 225
75−75−75=3×(−75)=−225.

19. Removing Zeros

Shortcut: Temporarily ignore zeros, subtract, then adjust.

Example: 7000 − 300 = ( 7 − 0.3 ) × 1000 = 67007000−300=
(7−0.3)×1000=6700.

CATEGORY 9: Advanced Patterns

20. Subtraction Using Ratios

Shortcut: Reduce ratios before subtracting.

Example: 45 : 60 − 3 : 4 = 3 : 4 − 3 : 4 = 045:60−3:4=3:4−3:4=0.

21. Subtracting Mixed Numbers

Shortcut: Convert to improper fractions before subtracting.

1 2 5
Example: 3 − 1 = ( 7/2 − 5/3 ) = 11/6 = 1 321​−132​=
2 3 6
(7/2−5/3)=11/6=165​.

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same for square(^2), cube(^3), ....

Here are the fastest techniques for finding squares and cubes (and higher
powers) in competitive exams:

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CATEGORY 1: SQUARING NUMBERS (x²)

1. Numbers Ending in 5

Shortcut: Use ( 𝑛 × ( 𝑛 + 1 ) ) × 100 + 25(n×(n+1))×100+25.

2
Example: 35 = ( 3 × 4 ) × 100 + 25 = 1225352=(3×4)×100+25=1225.

2. Squaring Two-Digit Numbers (a+b)

2 2 2
Shortcut: Use ( 𝑎 + 𝑏 ) = 𝑎 + 2𝑎𝑏 + 𝑏 (a+b)2=a2+2ab+b2.
2 2
Example: 43 = ( 40 + 3 ) = 1600 + 240 + 9 = 1849432=
(40+3)2=1600+240+9=1849.

3. Numbers Near a Base (10, 100, etc.)

Shortcut: Use
2 2 2
( 𝑏𝑎𝑠𝑒 + 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 ) = 𝑏𝑎𝑠𝑒 + 2 × 𝑏𝑎𝑠𝑒 × 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 + 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
(base+deviation)2=base2+2×base×deviation+deviation2.
2 2 2
Example: 102 = 100 + 2 × 100 × 2 + 2 = 10404
1022=1002+2×100×2+22=10404.

4. Difference of Squares
2 2
Shortcut: For ( 𝑥 − 𝑦 ) (x2−y2), use ( 𝑥 + 𝑦 ) ( 𝑥 − 𝑦 ) (x+y)(x−y).
2 2
Example: 25 − 24 = ( 25 + 24 ) ( 25 − 24 ) = 49252−242=(25+24)
(25−24)=49.

CATEGORY 2: CUBING NUMBERS (x³)

5. Numbers Ending in 5
3 2
Shortcut: Use 𝑛 = ( 𝑛 ) ( 𝑛 + 1 ) n3=(n2)(n+1).
3
Example: 15 = 225 × 16 = 3600153=225×16=3600.

6. Numbers Close to a Base

Shortcut: Use
3 3 2 2 3
( 𝑏𝑎𝑠𝑒 + 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 ) = 𝑏𝑎𝑠𝑒 + 3 × 𝑏𝑎𝑠𝑒 × 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 + 3 × 𝑏𝑎𝑠𝑒 × 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 + 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
(base+deviation)3=base3+3×base2×deviation+3×base×deviation2+deviation3.
3 3 2
Example: 102 = 100 + 3 ( 100 ) ( 2 ) + 3 ( 100 ) ( 4 ) + 8 = 1061208
1023=1003+3(1002)(2)+3(100)(4)+8=1061208.

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7. Binomial Expansion for Cubes

3 3 2 2 3
Shortcut: For ( 𝑎 + 𝑏 ) = 𝑎 + 3𝑎 𝑏 + 3𝑎𝑏 + 𝑏 (a+b)3=a3+3a2b+3ab2+b3.
3 3
Example: 12 = ( 10 + 2 ) = 1000 + 600 + 120 + 8 = 1728123=
(10+2)3=1000+600+120+8=1728.

8. Perfect Cubes Check

Shortcut: Compare the number with nearby perfect cubes.
3
Example: Is 512512 a perfect cube? Yes, 8 = 51283=512.

CATEGORY 3: HIGHER POWERS (x⁴, x⁵, etc.)

9. Fourth Power (x⁴)

2 2
Shortcut: Use the square of squares formula: ( 𝑎 ) (a2)2.
4 2 2 2
Example: 12 = ( 12 ) = 144 = 20736124=(122)2=1442=20736.

10. Fifth Power (x⁵)

5 3 2
Shortcut: Use 𝑥 = ( 𝑥 ) ( 𝑥 ) x5=(x3)(x2).
5 3 2
Example: 3 = ( 3 ) ( 3 ) = 27 × 9 = 24335=(33)(32)=27×9=243.

CATEGORY 4: SPECIAL TECHNIQUES

11. Repeated Digits

2
Shortcut: For ( 111 ) (111)2, use 1 × 12321 = 123211×12321=12321.
2
Example: 111 = 123211112=12321.

12. Square Roots

Shortcut: Approximate between perfect squares and adjust.
Example: 50 ≈ 49 + 1 = 7.150 ​≈49+1 ​=7.1.
√ √

13. Cube Roots

Shortcut: Memorize cubes of 1–10 and use interpolation.

Example: 3 1331 = 1131331 ​=11.
√

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for Square roots, cube root and .....

Here are the fastest shortcuts for calculating square roots, cube roots, and higher
roots for competitive exams:

CATEGORY 1: SQUARE ROOTS (√x)

1. Square Roots of Perfect Squares

Shortcut: Memorize the squares of numbers from 1 to 30 (or higher) to quickly
find square roots.
Example: 144 = 12144 ​=12, 169 = 13169 ​=13.
√ √

2. Square Roots of Non-Perfect Squares (Approximation)

Shortcut: Find the two nearest perfect squares and estimate the value.
Example: 5050 ​is between 49 = 749 ​=7 and 64 = 864 ​=8,
√ √ √
so 50 ≈ 7.150 ​≈7.1.
√

3. Estimating Square Root

Shortcut: Use the method of averaging.
Example: To find 5050 ​, guess between 7 and 8:
√
2
7.1 = 50.417.12=50.41, so 50 ≈ 7.0750 ​≈7.07.
√

4. Subtraction Method for Square Roots

2
Shortcut: If you know 𝑥 x2, subtract successive odd numbers to find the square
root.
Example: For 625625:
Start with 625 − 1 = 624625−1=624, 624 − 3 = 621624−3=621, and
continue subtracting odd numbers.
625625 requires 25 steps, so 625 = 25625 ​=25.
√

5. Square Root by Prime Factorization

Shortcut: Break the number into prime factors and pair them. The square root is
the product of the paired factors.
4 2 2
Example: 144 = √2 × 3 = 2 × 3 = 12144 ​=24×32 ​
√
=22×3=12.

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CATEGORY 2: CUBE ROOTS (³√x)

6. Cube Roots of Perfect Cubes

Shortcut: Memorize the cubes of numbers from 1 to 10.
Example: 3 27 = 3327 ​=3, 3 64 = 4364 ​=4.
√ √

7. Cube Root of Non-Perfect Cubes (Approximation)

Shortcut: Find the two nearest perfect cubes and estimate the value.
Example: 3 50350 ​is between 3 27 = 3327 ​=3 and 3 64 = 4364
√ √ √
​=4, so 3
50 ≈ 3.6350 ​≈3.6.
√

8. Estimating Cube Root

Shortcut: Use trial and error method or interpolation to estimate cube roots.
3
Example: 3
1283128 ​is between 5 and 6. You can try 5.1 = 132.651
√
5.13=132.651, which is close to 128, so 3
128 ≈ 5.043128 ​≈5.04.
√

9. Cube Root by Prime Factorization

Shortcut: Find the cube of prime factors and divide them into triplets.
3 9 3
Example: 3
512 = √2 = 2 = 83512 ​=329 ​=23=8.
√

CATEGORY 3: FOURTH ROOTS (⁴√x)

10. Fourth Root of Perfect Fourth Powers

Shortcut: Memorize the fourth powers of small numbers.

Example: 4 16 = 2416 ​=2, 4 81 = 3481 ​=3.
√ √

11. Fourth Root Approximation

Shortcut: Use interpolation between the two nearest perfect fourth powers.
Example: 4 1004100 ​is between 4 81 = 3481 ​=3 and 4 256 = 4
√ √ √
4256 ​=4, so 4
100 ≈ 3.24100 ​≈3.2.
√

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CATEGORY 4: HIGHER ROOTS (⁵√x, ⁶√x, etc.)

12. Fifth Root (⁵√x)

Shortcut: For numbers close to a known fifth power, estimate by finding the
nearest fifth power and refining.
Example: 5 32 = 2532 ​=2, 5 243 = 35243 ​=3.
√ √

13. Sixth Root (⁶√x)

Shortcut: Similar to fifth root, use trial and error between known sixth powers.
Example: 6 64 = 2664 ​=2, 6 729 = 36729 ​=3.
√ √

CATEGORY 5: ROOTS of Decimal Numbers

14. Roots of Decimals

Shortcut: Take the square or cube root of the decimal, then multiply by the
appropriate power of 10.
Example: 0.04 = 0.20.04 ​=0.2, 3 0.125 = 0.530.125 ​=0.5.
√ √

CATEGORY 6: USING VEDIC MATH (for Square Roots &

Cube Roots)

15. Vedic Method for Square Root

Shortcut: Use the "duplex method" to find square roots quickly.
Example: To find 57765776 ​, group the number in pairs from right to
√
left: 57∣7657∣76.
Start with the first group (57), find the closest square (7^2 = 49),
subtract 4949 from 5757, and bring down the next group (76).
Continue with the process to find the final answer: 5776 = 765776
√
​=76.

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16. Vedic Method for Cube Roots

Shortcut: Use the "digit-by-digit" method for cube roots.

Example: To find 3 337533375 ​, estimate by checking cube values
√
close to the number:
3
Since 15 = 3375153=3375, you can directly conclude that
3
3375 = 1533375 ​=15.
√

CATEGORY 7: PRACTICAL EXAMPLES

17. Square Root of Non-Perfect Squares (Quick Approximation)

Example: 3737 ​. Find the nearest perfect square (36).

√
Approximate 3737 as 6 + ( 37 − 36 ) /26+(37−36)/2 which gives 6.56.5.

18. Cube Root of Non-Perfect Cubes (Quick Approximation)

Example: 3
58358 ​. The cube roots of 27 and 64 are 3 and 4, respectively.
√
Approximate 5858 as 3 + ( 58 − 54 ) /23+(58−54)/2 which gives 3.23.2.

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