Square Root and Cube Root (Tips for Short Cuts)
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Chapter Contents |
· Introduction
· Square and Square Root of 2digit number whose unit place digit is 5
· Square and Square Root of any number
· Trick for Multiplication of two digit numbers ( unit place is 5 and difference of 1 in tens place digit)
· Trick for Multiplication of two digit numbers
· Trick for Multiplication of number with Base 10, 100, 1000……(close to 10, 100, 1000,…)
· Trick to Multiply by 25 and 125
· Trick for square of repetition of digit (1111, 222222, 33333,…..)
· Cube and Cube Root
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Square: when a number is multiplied by itself, then the product is called the square of the number.
For example: 64 is the square of 8 as 8 x 8 = 64, 144 is the square of 12, 100 is the square of 10 etc.
Square Root: The square root of a given number means such a number which when multiplied by itself, equals the given number.
For example: square root of 64 is 8 as 8 x 8 = 64, i.e. 64 = 8, square root of 144 is 12, square root of 100 is 10, etc.
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A. How to find Square and Square roots of Two Digit numbers whose unit place is 5 quickly |
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Trick 1 Square of two digit number whose unit place is 5 Example: 152 , 252, 352, 452, 552, 652, 752, 852, 952 |
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Rule : – 1. Write square of unit place digit = 52 = 25 (always) 2. now multiply 1 more than tens digit to tens digit and write it left of the 25 (Tens digit +1) x Tens digit /25 |
Do Yourself
a) 152 = (1 + 1) x 1/25 = 225
b) 252 = (2+1) x 2/25 = 625
c) 852 = ( 8+ 1 ) x 8/25 = 7225
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Trick 2 Square of two digit number whose unit place is 5 Example: 1052, 1152, |
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Rule : – 1. Write square of unit place digit = 52 = 25 (always) 2. Now multiply 1 more than tens digit to tens digit and write it left of the 25 (Tens digit +1) x Tens digit /25 |
Shortcut Method 1. 1052 = (10 +1) x 10/25 = 11025 2. 1552 = (15 + 1) x 15/25 = 24025
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Do Yourself
1252, 1352 , 1452, 1552 , 1652, 1752 , 1852, 1952
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B. How to find Square and Square roots of any Two Digit numbers Quickly |
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Trick 3 Square of any two digit number (including two digits whose unit place is 5) Example: 232, 252, 362 and other two digit number. |
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Rule:- Step1. Square of Unit Place (find the square of unit place and write only unit digit , add carry to 2nd step 2 Step 2. Unit digit x tens digit x 2 + Carry Step 3. Square of tens digit + Carry |
Shortcut Method 232 = 529 Step 1: 32 = 9 Step 2: 3 x 2 x 2 =12 = 1/2 = 2 ( use 1 as carry and to step 3) Step 3: 22 + 1 (carry from 2nd step) = 4 + 1 = 5 Ans: 529 |
Do Yourself
a) Find Value of 252, 362
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C. How to multiply when unit place is 5 and difference of 1 in tens place digit |
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Trick 4 Multiplication of two digit no. when unit place is 5 and difference in tens place is 1 Example: 35 x 45 |
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Rule: Step 1: Write 75 first Step 2: (Greater tens place digit + 1) x smaller tens place digit/75 |
Shortcut Method: 35 x 45 = 1575 Here , 4 is bigger digit and 3 is smaller digit 35 x 45 = (4 + 1) x 3/75 = 1575
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Example: 65 x 55
Solution: (6 + 1) x 5/75 = 3075
Do Yourself
85 x 95, 105 x 115
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D. How to Multiply two digit numbers |
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Trick 5 Multiplication of any two digit number Example: 56 x 65 |
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Rule:-
a b c d ———- Step 1:- b x d Step 2:- a x d + b + c + Carry Step 3:- a x c + Carry |
Shortcut Method
5 6 6 5 ———- Step 1:- 6 x 5 = 30 = 0 (3 is carry) Step 2:- 5 x 5 + 6 x 6 + 3 (carry) = 64 =4 ( now 6 is Carry) Step 3:- 5 x 6 + 6 (Carry) = 36 Ans.3640 |
Do Yourself
(a) 69 x 81
(b) 88 x 83
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E. How to multiply two digits when base is 10 |
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Trick 6 Multiplication of any two digit number < 20 (Base is 10) Example: 14 x 12 |
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Rule:
Step 1: How much 1st number more than 10, write with + sign a = B + p , where B=10 Step 2: Repeat this step 1 with 2nd number b = B + q , where B= 10 Step 3: a +p
x b +q here , a + q = b + p = K (always same) Now , Step 4: Calculate (p x q) and write on right side (only one digit make other digit carry (c) )
Step 5: Calculate a + q or b + p (both will be same and write it on left side with addition of carry(c) Ans: (a + q) + c (if any) /(p x q)
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Shortcut Method
Ex :- 14 x 12 Step 1:- 14 = 10 + 4 Step 2:- Repeat this step 1 with 2nd number 12 = 10 + 2 Step 3:
14 +4
X
12 +2
Note : 14 + 2 =12 + 4 = 16 ( total will always will be same) Step 4: ( 4 x 2) =8 Step 5: (14 + 2 =16 = 12 + 4) Ans: (14 + 2 + 0 (no carry)/(4 x 2) = 168
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Example: 16 x 19
Solution: 16 +6
X
19 +9
——————–
(16 + 9) = 25
6 x 9 =54 = 4 and 5 = c (carry)
Ans. (16 + 9) + 4 /5 = (29 +4) /5 = 335
Do yourself
(a) 18 x 17
(b) 14 x 19
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Trick 7 Multiplication when one number more than 20 Example: 25 x 18 |
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25 + 15
X
18 + 8
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Step 1: 15 x 8 = 120 = 0 ( 12 is carry)
Step 2: 25 + 8 = 18 + 15 = 33
Step 3: 33 + 12 (carry)/0
= 450 |
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Trick 8 When one number is less than 10 Example: 28 X 8 |
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A = 10 + p B = 10 – q
a +p X b – q
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Short cut Method 1 Step1: (p X – q ) = – pq = p and q Unit Place : 10 – q = R Step 2: (a – q) = (b + p) = K Left place = K- p = L Ans : L/R
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Shortcut Method 2 28 +18 X 8 – 2 Step 1:- 18 x -2 = – 36 = 3 and 6 Unit place = 10 – 6 = 4 Left digit = (28 – 2) = ( 8 + 18) =26 26 – 3 = 23
Ans: 234 Note:- better we should multiply it orally |
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Trick 9 Multiplication when base is 100 for two Digits < 100 Example: 98 x 97 |
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Rule
98 -02 X 97 -03
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Shortcut Method
Base is 100 Step1: 98 – 100 = -02 97 – 100 = -03 Step 2:- 02 x 03 = 06 , write on right side use only two digit 3rd would be Carry Carry = 0 Step3:- 98 -03 = 97 -02 = 95 + Carry if any , write on left side Ans :- left side digit + carry / Right side digit = 95 + 0 /06 |
Example: 89 x 87
Solution 89 -11
X
87 -13
Step1: -11 X -13 = 143 = 43 (1= carry)
Step2: 89 – 13 = 87 – 11 = 76
Ans: 76 + carry /43 = 76 + 1 /43 = 7743
Note: We can also find the square of any dwo digits close to 100
Do Yourself
a) 89 x 87
b) 86 x 94
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Trick 10 Multiplication when base is 100 for two digit > 100 Example: 104 x 106 |
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Rule 104 +04 X 106 +06 ___________ (104 + 6)/(6 x 4) = 110/024 = (110 + 0)/ 24 = 11024 |
Shortcut Method Base 100 Step1: 104 – 100 = +04 106– 100 = + 06 Step 2: (+ 04) x (+ 06) = 024 , write on right side use only two digit 3rd would be Carry Carry = 0 Step3: 104 + 4 = 106 + 04 = 110 Step 4: 110 + 0 ( Carry from 2nd step if any) , write on left side
Ans : left side digit + carry / Right side digit = (110 + 0) /24 Ans : 11024 |
Example: 112 x 112
Solution: 112 +12
X
112 +12
_____________
(112 + 12)/ (12 x 12)
= 124/144 = (124 + 1) / 44 = 12544 ( 1 = carry)
Ans: 12544
Note: We can also find the square of any three digits close to 100
Do Yourself
a) 112 x 112
b) 102 x 113
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Trick 11 Multiplication when base is 100 for two Digit one > 100 and other < 100 Example: 98 x 104 |
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Shortcut Method
98 -02
X
104 +04 ___________ (98 + 04)/(-02 x +04) =102 /-008 = (102 -1 + 0)/(100 – 08)= 10192 |
Base 100
Step 1: (-02) x (+04) = -008 = (0) –(08) Carry =0 Step 2: (100-08) on Right side Step 3: 98 + 04 = 104 – 02 = 102 Step 4: (102 – 1 + carry), on left side
Ans : (102 – 1 + 0 )/(100 – 08 ) = 101/92 = 10192 |
Example: 89 x 111
Solution: 89 -11
X
111 +11
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(89 + 11 – 1 + C)/-11x (+11)
= (99 + C)/ (100- 121)
= (99 -1)/ (100 – 21) = 9879
Here carry is -1 (we will break – 121 = -1 – 21 )
Do Yourself
a) 99 x 112
b) 106 x 94
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Trick 12 Square of a number Close to 100 (100 + a) Example: 1042 |
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Shortcut Method 1042 = (104 + 4)/ 42 = 10816
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Step :-1 100 + a Step:-2 (100 + a + a + c )/a2 Where c = carry on left side only two digit, make third digit as carry |
Example: 1122
Solution: (112 + 12)/ 122
= 112/144 = (112 + 1)/44( where c = 1)
= 113/44 = 11344
Do Yourself
a) 1122
b) 1132
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Trick 13 Square of a number Close to 100 (100 – a) Example: 942 |
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Shortcut Method 942 = (94 – 6)/ 62 = 88/36
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Step 1: 100 – a Step 2: (100 – a – a + c )/a2 Where c = carry On left side only two digit, make third digit as carry |
Example: 822
Solution: 822 = (82 – 18) + C /182 =( 64 + c )/324
= (64 + 3)/ 24, where c = 3
Ans: 6724
Do Yourself
1. 822
2. 782
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Trick 14 Square of a number Close to 1000 (1000 + a) Example: 10042 |
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Rule:- 10042 = (1004 + 4)/ 42 = 1008/016 Ans:- 1008016 |
Step :-1 1000 + a Step:-2 (1000 + a + a + c )/a2 Where c = carry On left side only three digit, make 4th digit as carry |
Do Yourself
a) 10122
b) 10252
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Trick 15 Square of a number Close to 1000 (1000 – a) Example: 9942 |
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Shortcut Method 9942 = (994 – 6)/ 62 = 988/036 Ans :-988036 |
Step :-1 1000 – a Step:-2 (1000– a – a + c )/a2 Where c = carry On left side only three digit, make 4th digit as carry |
Example: 9822
Solution = (982 – 18)/182 = 964/324
= (964 + 3)/24, where C = 3
Ans = 96724
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SQUARE OF TWO DIGIT NUMBERS
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F. How to multiply a number by 25 |
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Trick 16 Multiply any number by 25 Example: 48 x 25 |
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Shortcut Method
48 x 25 = 48004=1200 |
Step 1:- write two zero (00) to right side of other number Step 2 :- Divide resultant by 4 Ans :- no. x 1004 |
Do Yourself
a) 56728 x 25
b) 123432 x 25
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How to multiply a number by 125 |
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Trick 17 Multiply any number by 125 Example: 488 x 125 |
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Shortcut Method: 488 x 25 = 4880008=61000 |
Step 1: write three zero (000) to right side of other number Step 2 : Divide resultant by 8 Ans : Number x 1008 |
Do Yourself
a) 56728 x 125
b) 123432 x 125
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Trick 18 Square of repetition of 1 Example: (111)2 |
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Rule:- (111)2 = 12321 |
Step:- n =3 ( digit of 1) Step 2:- write ascending number from 1 to 3 and then descending order till 1 |
Example: (1111111)2
Solution: 1234567654321
Example: (11111)2
Solution: 123454321
Example: 222 + 2222 +22222+ 222222
Solution: 4 (12 + 112 + 1112 +11112 +111112)
= 4 (11 + 121 + 12321 + 1234321 + 123454321)
= 4 (124701084)
= 498804336
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Trick 19 Square of repetition of 3 Example: (33)2 |
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Shortcut Method ( OZEN) In case of 3 O – 1 (one two times) Z – 0 ( zero one time ) E – 8 ( eight two times ) N – 9 ( one times)
(333)2 = 110889 (33333)2 = 1111088889 |
Step 1 :- Count 3 in given figure = n (say) Step 2:- Write 1 n-itimes Step 3:- Write 0 ( one times only) Step 4: Write 8 n-itimes Step 5:- Write 9 ( one times only) |
Do Yourself
a) (333)2
b) (33333)2
c) 32 + 332 + 3332 +33332 +333332
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Trick 20 Square of repetition of 9 Example: (99)2 |
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Shortcut Method ( NEZO) In case of 9 N – 9 ( two times) E– 8( one time ) Z – 0( two times ) O – 1 ( one times) (999)2 = 998001 (99999)2 = 9999800001 |
Step 1 :- Count 9 in given figure = n (say) Step 2:- Write 9 n-itimes Step 3:- Write 8 ( one times only) Step 4: Write 0n-itimes Step 5:- Write 1 ( one times only) |
Do Yourself
a) (9999)2
b) (99999)2
c) 92 + 992 + 9992 +99992 +999992
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Trick 21 Square of repetition of 6 Example: (66)2 |
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Shortcut Method In case of 6 Fo– 4 ( two times) Th– 3 ( one time ) Fi– 5( two times ) Si – 6 ( one times) (666)2 = 443556 (99999)2 = 4444355556 |
Step 1 :- Count 6 in given figure = n (say) Step 2:- Write 4 n-itimes Step 3:- Write 3 ( one times only) Step 4: Write 5n-itimes Step 5:- Write 6 ( one times only) |
Do Yourself
a) (6666)2
b) (666666)2
c) 62 + 662 + 6662 +66662 +666662
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Trick 22 Multiplication of any two digit number Example: 27 x 52 |
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Shortcut Method a b
c d ____________ ac + C2/(ab + bc)+ C1 /bd where C1 and C2 are carry obtained by multiplying bd and ( ab + bc) |
Step 1:- 2 7
5 2 Step 2:- 7
2 = 7 x 2 = 17 = 4 , here (1 = C1) Step 3:- 2 7
5 2
= ( 2 x 2 ) + ( 5 x 7) + C1 Put C1 = 4 + 35 + 1 = 40 = 0 ,here C2 = 4 Step 4:- 2
5 = 2 x 5 + C2 = 10 + 4 = 14 Ans :- 1404 |
Do Yourself
a) 38 x 47
b) 86 x 34
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Trick 23 Multiplication of three digit number Example: 234 x 432 |
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Shortcut Method
a b p
c d q ________________________________________ (ac + C4)/(ad + bc + C3 )/(aq + pc + bd + C2) / (bq + pd + C1 )/ bd
where C1 , C2 , C3 and C4 are carry obtained by multiplying pq , (bq + pd), ( aq + pc + bd ), (ad + bc) and ac respectively. |
Step 1: 2 3 4
4 3 2
Step 2:
4
2 = 4 X 2 = 08 = 8 here, C1 = 0
Step 3:- 3 4
3 2 = (3 x 2) + (3 x 4) + C1 = 6 + 12 + 0 = 18 = 8 , here C2 = 1
Step 4:-
2 3 4
4 3 2 = (2 x 2) + (4 x 4 ) + ( 3 x 3) + C2 = 4 + 16 + 9 + 1 = 30 = 0 here, C3= 3
Step 5:- 2 3
4 3 = (2 x 3) + ( 4 x 3 ) + C3 = 6 + 12 + 3 = 21 = 1 here, C4 =2
Step 6:- 2
4 =( 2 x 4 ) + C4 = 8 + 2 = 10
Ans :- 101088 |
Do Yourself
a) 534 x 124
b) 123 x 675
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Trick 24 Multiplication of any number with 9 and its recitative no. (9, 99, 999, 9999 etc.) Example: 8653 X 9 |
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Shortcut Method Step 1:- count no of 9 Step 2:- put same no. of zero on right of other figure Step 3:- Now , Stract the original no. from new no. obtained |
8653 x 9 Step 1:- no. of 9 =1 Step 2:- put one 0 on right side of the other no. 86530 Step 3:- 86530 -8653 ______ 77877 |
Example: 8932 x 99
Solution: 893200
– 8932
___________
884268
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G. How to Calculate Multiplication of two no. made of repetition of same no. |
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Trick 25 Multiplication of two no. made of repetition of 1 Example: 11111 x 1111
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Shortcut Method n x m = digit of multiple x digit of multiplier where m < n Step 1:- write in ascending starting from 1 upto m Step 2:- write m, (n – m + 1 )times Step 3:-write in descending from m -1 to 1
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11111 x 1111 n = 5, m =4 Step1:- 1234 Step 2:- ( 5 – 4 + 1)times = 2 times =12344 Step 3:- 12344321 |
Example: 111111 x 111
Solution: n = 6 , m =3
Write 3, (6 -3 + 1)times = 4 times
Ans. 12333321
Example: 2222 x 222 =
Solution: 22 (1111 x 111) = 4(123321) =482394
