CUBE

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                                                Cube

Method 1:We will explain with a exmple.

Ex:Find the cube of 51 i.e. 51^3

Step 1:Cube ten's digit that means 5^3

125

Step 2: Compare the ratio between Ten's Digit and Unit's Digit that means compare 5 ratio with 1 i.e. 5:1

Now we will write next digit to right hand side with ratio 5:1

125                25             5                 1 ------------First Row

Step 3:Now Double the digit 2nd and 3rd from right hand side

                     50            10                     ----------Second Row

Step 4:Add the First and Second Row

125               25              5                  1

                     50             10

125               75             15                  1

132(Add Carry 7) 6(Carry 7) 5(Carrry 1 ) 1

i.e

51^3 = 132651

Method 2:We will explain with a exmple.

Ex:Find the cube of 87 i.e. 87^3

First(Ten) Digit =8

Second (Unit)Digit=7

Step 1:

First Row::Write down the Ten's Digit Cube Square Digit

8^3               8^2                  8

512                64                    8

Second Row::Write down the unit's Digit Digit Square Cube from leave one place from left handside.

                            7                    7^2              7^3

                       7                    49                 343

Third Row::Multiply 2nd and 3rd place of First row and Second row Then Add First Row and Second Row.

512                      64                        8

                            *7                       *49                  343

512                     448                      392                 343

Four Row:Multiply by 3 (Because we are doing cube) in the third row at 2nd and 3rd place digit.

512                   448                      392                    343

                           *3                        *3

512                   1344                    1176                  343

658(Add 146) 5(Add 121 carry 146) 0(Add 34 and Carry 121) 3(carry 34)

i;e

87^3=658503

Square Root & Cube Root

                                          Square Root and Cube Root

  Shortcut Trick use for Perfect Square Root or Cube Root

Number	Square	Last Digit of its Square Root	Cube	Last Digit of its Cube Root
1	1	1	1	1
2	4	4	8	8
3	9	9	27	7
4	16	6	64	4
5	25	5	125	5
6	36	6	216	6
7	49	9	343	3
8	64	4	512	2
9	81	1	729	9
10	100	0	1000	0

Observation

1.If the unit digit of any number is 2 or 8 then its square will have unit digit as 4.

2.If the unit digit of any number is 3 or 7 then its square will have unit digit as 9.

3.If the unit digit of any number is 4 or 6 then its square will have unit digit as 6.

4.If the unit digit of any number is 1 or 9 then its square will have unit digit as 1.

5.If the unit digit of any number is 0 then its square will have unit digit as 0.

Example

1.Find the square root of 529

5,29(Last two digit)

3..............9

7..............9

3/7

5

5 is between 2 & 3

So value is 2

2 * 3=6

As compare 5 with 6 ,So 5 is less than 6 so value of unit is 3(i.e Smaller)

Now, Square Root of 529 is 23

2.Find the square root of 6084

60,84(Last two digit)

2..............4

8..............4

2/8

60

60 is between 49 & 64

So value is 7

7 * 8=56

As compare 56 with 60 ,So 60 is large than 56 so value of unit is 8(i.e Larger unit digit)

Now, Square Root of 529 is 78

Square Root

1. First Find the last digit of the square root ,which can directly be obtained by looking the last digit of the number and then referring the table.
2. Next,Ignore the last 2 digits of the number and look at the number which remain. Think of a number ,whose square is just equal or less then this remaining number.

Ex:Find the Square Root of 10 24.

Last Digit of 1024 is 4

According to Above Table

2........................................................4

8........................................................4

Ignore the Last two digit that is 24

Now Focus on 10

Now find the square of number which is equal or lesser than 10.

3^2=9

4^2=16

From here we find square of 3 is less than to 10.

So,This is the ten's digit of square root is 3

Now 9 (Smaller than 10 ,so we will choose small digit i.e. 2) is less than 10 that why unit digit is 2.

Now ,Answer of square root of 1024 is 32.

Cube Root

This technique helps you find out the cube root of a 4 or 5 or 6 digits number mentally.

Note:

1. Cube of a 2-digit number will have at max 6 digits (88^3 = 681,472). That implies if you are given with a 6 digit number, its cube root will have 2 digits.
2. This trick works only for perfect cubes, it will not work for any arbitrary 6-digit
3. It works only for integers

Example: 

Say you have to find the cube root of 166375. It is known that it’s a perfect cube. 

Step 1: 

Now divide this number into two parts. The right hand side should always have 3 digits. Remaining digits will
come in left hand side. Do it as shown below.
166.........................................375

You know the answer will have 2 digits. Digit at tens place and digit at units place. We will get the digit at tens place using the left hand side of the original number (166) and digit at units place using right hand side of the number (375)

Step 2.
For left hand side we need to use Above Table . We have to see the above table in which 166 lies between 125 and 216. So we will take the cube root of the smaller number i.e. 125 which is 5.

So 5 is the tens digit of the answer.
Step 3.
For right hand side we need to use Above Table. Since our original number (the perfect cube) ends in 5 (see 166375), its cube root will ends in 5.
Thus the units digit will be 5.

Step 4:
Combining the results we get the answer as 55.

(166375)^1/3 = 55

MULTIPLICATION

                                  Multiplication Trick

Method 1 :Multiplication of 2 Digit Numbers Where the first digit of both numbers is same and Last digit of the 2 Numbers sum to 10.
(1)56*54
First Digit of 56 is 5
First Digit of 54 is 5


Last Digit of 56 is 6
Last Digit Of 54 is 4
Sum of Last Two digit is 6+4=10



Multiply of 5 and (5+1)
5*(5+1)=5*6=30 Write Down 30
Multiply of Last Digit of Both Numbers
6*4=24 write Down 24
i.e
56*54=3024

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                                                                    OR

Multiplication of 2 Digit Number the Ten's Digit Add up to 10 and Unit digits Same

1.First Find the multiplication of last two digit of both the numbers

2.Multiply the Ten's Digits and Add the Common Digit to the multiplication

Ex:44 * 64

First Number is 44

First Digit of 44 is 4

Second Digit of 44 is 4

Second Number is 64

First Digit of 64 is 6

Second Digit of 64 is 4

Second Digit in Both Number is Same i.e 4

Sum of First Digit of Both Number is 4+6=10

Multiplying the unit Digit ,we get 4*4=16,Put it at right hand side.

Multiplying the Ten's Digits of Both Number and Adding Common Digit ,we get (4*6) + 4=24+4=28 ,Put it at the left hand side.

So,We get Answer is 2816

Method 2 :Multiplication of two digit number that differ by 6
(1)10*16
First Number =10
Second Number =16
Difference Between 10 & 16 is 16-10=6
If two number differ by 6 then their product is the square of  their average  minus 9 .
Average of 10 and 16 is (10+16)/2=26/2=13
Square of Average Number
13^2 =169
Minus 9 from Square of average
169-9=160
i.e
10*16=160



Method 3 :Multiplication of two digit number that differ by 4
(1)10*14
First Number =10
Second Number =14
Difference Between 10 & 14 is 14-10=4
If two number differ by 6 then their product is the square of  their average  minus 4 .
Average of 10 and 14 is (10+14)/2=24/2=12
Square of Average Number
12^2 =144
Minus 4 from Square of average
144-4=140
i.e
10*14=140

Method 4 :Multiplication of two digit number that differ by2
(1)10*12
First Number =10
Second Number =12
Difference Between 10 & 12 is 12-10=2
If two number differ by 6 then their product is the square of  their average  minus 1 .
Average of 10 and 12 is (10+12)/2=22/2=11
Square of Average Number
11^2 =121
Minus 1 from Square of average
121-1=120
i.e
10*12=120
Method 5 :Multiplication of 125 with Any Number.
(1)92*125
125=5^3=(10/2)^3=1000/8
Now,
92*125
92*(1000/8)
(92*1000)/8
92000/8
11500
i.e
92*125=11500

Method 6 :Multiplication of 25 with Any Number.
(1)92*25
25=5^2=(10/2)^2=100/4
Now,
92*25
92*(100/4)
(92*100)/4
9200/4
2300
i.e
92*25=2300

Method 7 :Multiplication of 5 with Any Number.
(1)92*5
5=5^1=(10/2)^1=10/2
Now,
92*5
92*(10/2)
(92*10)/2
920/2
460
i.e
92*5=460


Method 8 :Multiplication of 999 with Any Number.
(1)92*999
999=1000-1
Now,
92*999
92*(1000-1)
(92*1000)-(92*1)
92000-92
91908
i.e
92*999=91908

Method 9 :Multiplication of 99 with Any Number.
(1)92*99
99=100-1
Now,
92*99
92*(100-1)
(92*100)-(92*1)
9200-92
9108
i.e
92*99=9108


Method 10 :Multiplication of 9 with Any Number.
(1)92*99
9=10-1
Now,
92*9
92*(10-1)
(92*10)-(92*1)
920-92
828
i.e
92*9=828

Method 11 :Multiplication of 11 with Any Number.
(1)92*11
Write down 0 at the both end of Number(it is called Zero Sandwich )
0920
sum of the two successive number digit
0+9....9+2....2+0
9...11....2
10.....1(carry 1)....2

1012 
i.e92*11=1012
                                     OR
Insert the sum of 2 successive digits and to put 2 terminal digit in its place

456872*11
Put Two Terminal Digits in its place
4.........................................................................2

Insert the Sum of 2 Successive digit
4....(4+5).....(5+6)....(6+8)....(8+7).....(7+2).......2

4.....9......11.......14......15.....9.....2
 4.....9+1(Take Carry 1)......11-10(10 Carry Forwarded to left hand-side)+1(Carry take 1 from Right hand side).....14-10(10 Carry Forwarded to left hand-side)+1......15-10(10 Carry Forwarded to left hand-side).....9....2
4...........10.............2..............4+1...............5.........9...........2
4+1........10-10...........2...............5................5.........9..........2
 5..................0..............2...........5.......5...........9..........2
5025592
 i.e 
456872*11=5025592
Note: if sum of two digit is greater than 10 then add 1 to previous digit and subtract 10 to the associate digit



Method 12 :Multiplication of 12 with Any Number.
(1)65214*12
Write down 0 at the both end of Number(it is called Zero Sandwich )
0652140
The Ultimate Digit is 0 and the Penultimate digit is 4 .
Adding the ultimate digit and twice the penultimate digit we get 0+8=8

For the Ten's Column ,The ultimate digit is 4 and the Penultimate digit is 1 so 4+2=6

Like wise 1+4=5 and 2+10=12 ,with 12 We set down 2 and carry 1.
5+12+carry1=18 and again we carry 1.

The Final step is 6+0+carry 1=7

So ,the Answer is 782568

Method 13 :Multiplication of Any two Number, Both Ranging between 11 to 19.

(1)13*19
Add First Number and Last digit of second number Take 0 (Zero) in the third place of this number the add product of last digit of the two numbers in it.

First Number is 13
Second Number is 19

Add First Number and Last Digit of Second Number
First Number is 13
Second digit of second number is 9
13+9=22

Put 0 (Zero)
0

Product of Last Digit of two Number.
Last Digit of 13 is 3
Last digit of 19 is 9
3*9=27

Add Product of Last Digit of two Number in Number
22.....0 + 27
220+27
447
i.e.
13*19=447

SQUARE

                        Square of 2 Digit

Method 1:If unit digit is 5 then,
(25)^2
That Means

2 5
Add 1
2+1(Add 1 More)=3

Square of unit Digit
5^2 = 25

Now
2*3=6

(25)^2=625

Method 2:Let me explain this trick by taking examples
67^2 = [6^2] [7^2] + 20*6*7 = 3649+840 = 4489
similarly
25^2 = [2^2][5^2]+20*2*5 = 425+200 = 625

Method 3:Square of any 2 digit number

Let me explain this rule by taking examples
27^2 = (27+3)*(27-3) + 3^2 = 30*24 + 9 = 720+9 = 729
In this method, we have to make a number ending with 0, that's why; we add 3 to 27.

1 more example
78^2 = (78+2)*(78-2) + 2^2 = 80*76 + 4 = 6080+4 = 6084

Method 4:Square of any 2 digit number

The Method is to use (a+b)^2 Formula in the following Manner a^2 / 2ab /b^2

(1)85^2

8 5 Break in two part 8 & 5

8^2....... 2*8*5......... 5^2 (square of 8 / 2 * 8 *5 / square of 5)

64...............80...............25

64..............82................5(Carry 2)

72..............2(Carry 8).............5

7225

i.e

85^2=7225

Method 5:Square of numbers near to 50
Let me explain this rule by taking examples
(1)37^2 :-
37 is the near by 50 so, First Calculate 50-37 ,it is 13
13 is less than 50 ,Therefore deduct 13 from 25 i.e. 25-13(How Much Less)
25-13=12
13^2=169
Now
12.....169 (carry 1 )

37^2=1369

(2)57^2 :-
57 is the near by 50 so, First Calculate 50+7(How Much More).
7 is greater than 50 therefore Add 7 to 25 i.e. 25+7
25+7=32
7^2=49
Now
32.....49 (No carry )

57^2=3249

Method 6:Square of numbers near to 100
Let me explain this rule by taking examples
(1)96^2 :-
First calculate 100-96, it is 4(How Much Less)

4 is less than 100 ,Therefore deduct 4 from 96 i.e. 96-4=92
4^2=16
We should written 16 As 16
Now
92.....16 (No carry )

96^2=9216
(2)115^2
115 is the near by 100 so, First Calculate 100+15(How Much More).  15 is greater than 100 therefore Add 15 to 115 i.e. 115+15
115+15=130
15^2=225
Now
130.....225 (Carry 2)

115^2=13225

           Square of Containing Repeated Number

Method 1 :Repetition of 1

Step 1:Count the digit,Count=N

Step 2:Now starting From 1 Write The Number till N

Step 3 :Now Starting From N Write number till 1

Ex: 11111^2

First we See that there are 5 times 1

Now we Write number from 1 to 5

Now Again From From 5 to 1.

So Answer is 123454321

Method 2 :Repetition of 9

Step 1:Count the digit,Count=N

Step 2:First Write N-1 times 9 then 8

Step 3 :Again N-1 times 0 then 1

Ex:9999^2

We See that there are 4 times 9

Now we write 4-1=3 times 9 then 8

9998

Now 3 Times 0 then 1

0001

i.e

9999^2=99980001

Method 3 :Repetition of 3

Step 1:Count the digit,Count=N

Step 2:First Write N-1 times 1 then 0

Step 3 :Again N-1 times 8 then 9

Ex:333333^2

We See that there are 6 times 3

Now we write 6-1=5 times 1 then 0

111110

Now 5 Times 8 then 9

888889

i.e

333333^2=111110888889