Square Roots of Perfect Squares

Square root is the reverse for square. If you memorized the squares of 1 to 10, you can easily answer the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. How about the square roots of perfect squares greater than 100? Image below shows the relation between square and square root.

The method is done by reversing the process of squaring 2-digit numbers. Let's take an example from the image, square root of 169. Segregate the digits of 169:
1      6      9

First, think of a number that when squared will equal to 1 .
1*1 = 1
So, first digit of the answer is 1.
Next, double this number then divide from the next digit 6.
6/(2*1) = 3
So, last digit of the answer is 3.
Square root of 169 = + - 13  (positive and negative 13)

To check if perfect square, take the last digit 3 then square it.
3*3 = 9 This equals to the last digit of the problem 169. Therefore, perfect square.

More examples:

Square root of 1681 = ?
16     8          1
4*4   8/(2*4)
4     1
Square root of 1681 = + - 41

Square root of 441 = ?
4         4             1
2*2     4/(2*2)
2       1
Square root of 441 = + - 21

Square root of 3600 = ?
36       0      0
6*6      0
6       0
Square root of 3600 = + - 60

Square root of 1225 = ?
12        2         5

Here, 12 is not a perfect square. We will use a number that if squared will be near to 12, but not over it. If 3x3 = 9, less than 12. If 4x4 = 16, more than 12. Therefore, use 3, then there will be remainder of 12 - 3*3 = 3. Put this remainder beside the next digit 2 and become 32.

32/(2*3) = 5 with remainder 2

Square root of 1225 = + - 35

Rewriting the solution:

Square root of 1225 = ?
12        2                      5
n*n      R2/(2n)             r5  R = 12 - 3*3 = 3
3*3    32/(2*3)              r5
3        5 with r = 2       25
Square root of 1225 = + - 35
5*5 = 25 = r5   Therefore, perfect square.

Square root of 888 = ?
888 is not a perfect square. There is no perfect square ending with 2, 3, 7 and 8. This can be check on the squares of 1 to 10.

These are long to write but quick on the mind.

#Matamata

Square of Numbers Ending with 5

To find the square of any two digit number ending with 5, multiply the first digit by itself plus 1 then put 25 after it.

Example

35 x 35 = ?

First digit is 3 multiplied with 3+1 equal to 4.

3x4=12 , then put 25 after it.

35 x 35 = 1225

More examples

65 x 65 = ?

6x7 = 42

65 x 65 = 4225

75 x 75 = ?

7x8 = 56

75 x 75 = 5625

#Matamata

Cube Roots of Perfect Cubes

Calculate cube roots of perfect cubes in seconds.

Preparation:

1. Memorize the cubes of 1 to 10 shown below.

1³  =  1

2³  =  8

3³  =  27

4³  =  64

5³  = 125

6³  = 216

7³  = 343

8³  = 512

9³  = 729

10³ = 1000

2. From the above cubes of 1 to 10, remember the following properties.

If the last digit of the perfect cube = 1, 4, 5, 6, 9 & 0, the last digit of cube root is equal to that number.

If the last digit of the perfect cube = 2, 3, 7 & 8, the last digit of the cube root = 8, 7, 3 & 2 respectively, their  10’s complement.

It’s very easy to remember.

1  ->  1  (Same numbers)

8  ->  2 (8+2 = 10)

7  ->  3 (7+3 = 10)

4  ->  4 (Same numbers)

5  ->  5 (Same numbers)

6  ->  6 (Same numbers)

3  ->   7 (3+7 = 10)

2  ->   8  (2+8 = 10)

9  ->   9 (Same numbers)

0  ->   0 (Same numbers)

Take note also that

8 -> 2 and 2 ->  8 

7 -> 3 and 3-> 7

Example 1:  Find Cube Root of 4913

Step 1

Split the digits, group of 3 digits.

      4        913

Step 2

Take the first group which is 4 .

Find out a maximum cube that is not greater than 4. Use 1³ = 1.

Hence the first digit of the answer  = 1. 

Step 3

Take the last group which is 913. 

The last digit of 913 is 3.

Remember point 2, If the last digit of the perfect cube = 3, the last digit of the cube root = 7

Hence the last digit of the cube root  = 7

That is, the answer = 17

More examples:

Cube root of 804357 ?       

804           357

9³ = 729          7

9                     3

Answer: 93

Cube root of 9261 ?

9              261

2³ = 8              1

2                     1

Answer: 21

Cube root of  32768 ?

32               768

3³ = 27            8

3                     2

Answer: 32

Cube root of 97336 ?

97            336

4³ = 64            6

4                     6

Answer: 46

Cube root of 216000 ?

216          000

6³ = 216          0

6                     0

Answer: 60

It is as simple as that, all you need to do is memorize the cubes of 1 to 10.

#Matamata

Memorizing Tip

Memorizing tip for students:

During school days, studying is not hard as it is. Do what you want to do in your free time. At home, eat dinner, watch TV shows, play games and have fun. But before going to sleep, read your school notes without memorizing what you read. Be it Math, Science or History, just read and do not memorize. Every school day, new lessons, new notes, new chapter to read. Read from the very start until the present. Do it every night.

Eventually, you will be amazed that at time of your exams, you will remember what you read and even have memorized it already, without the intention of memorizing it in the first place. This technique will make you always prepared whenever there is an exam. You don't even have to study the night before exam. You become always ready for it. You will not be afraid anymore to be called and answer your teacher's questions about your lessons.

Just read, don't think or analyze, don't pressure your mind. Just read, read from start to end, every night. Feed your mind.


#Matamata

Simple Product of Big Numbers

Some people find it hard to multiply big numbers. There is a trick to do it simpler. This is applicable for numbers near a base. A base can be 10, 100, 1000 or higher.

Example:
98 x 94 = ?

This question can be answered in seconds. The answer is 9212.

Both 98 and 94 are numbers near 100. First, take their differences from 100.
100 - 98 = 2
100 - 94 = 6
Then, subtract one difference from the other number.
98 - 6 = 92
94 - 2 = 92
It will always have the same answer, so choose which one is easier for you. This is the first two digits of the final answer. The other two digits is the product of the differences 2 and 6.
2 x 6 = 12
Putting 92 & 12 together, we have 9212 as the answer.

 -2    -6
98 x 94 = ?
98 - 6 = 92 
2  x  6 = 12
98 x 94 = 9,212

More examples:

98 x 87 = ?

-2   -13
98 x 87 = ?
87 - 2 = 85
2 x 13 = 26
98 x 87 = 8,526

998 x 986 = ?

  -2    -14
998 x 986 = ?
986 - 2 = 984
2  x  14 = 028   (base is 1000, 3 zeros means 3 digits for answer)
998 x 986 = 984,028

7 x 8 = ? 

-3    -2
7  x  8 = ?
7  -  2 = 5
3  x  2 = 6   (base is 10, 1 zero means 1 digit for answer)
7  x 8 = 56

For numbers above the base, the trick becomes easier. Instead of subtracting the differences, add.

    3       5
103 x 105 = ?
103  +  5 = 108
3 x 5 = 15   (base is 100, 2 zeros means 2 digits for answer)
103 x 105 = 10,815

    7       8
107 x 108 = ?
107 + 8 = 115
7 x 8 = 56   (base is 100, 2 zeros means 2 digits for answer)
107 x 108 = 11,556

  3      5
13 x 15 = ?
13 + 5 = 18
3 x 5 = 15   (base is 10, 1 zero means 1 digit for answer. Therefore add 1 to 18, 18+1=19)
13 x 15 = 195

For numbers that one is below the base and the other one is above the base, subtract the base to these numbers and consider the positive and negative differences.

Example:
96 x 108 = ?

 -4     +8
96 x 108 = ?
96 + 8 = 104
108 - 4 = 104 (same answer)
(-4) x 8 = (-32) 
96 x 108 = 10,4(-32)
96 x 108 = 10,400 - 32
96 x 108 = 10,368

More examples:

98 x 114 = ?

 -2   +14
98 x 114 = ?
114 - 2 = 112
98 + 14 = 112 (same answer)
(-2) x 14 = (-28) 
98 x 114 = 11,2(-28)
98 x 114 = 11,200 - 28
98 x 114 = 11,172

102 x  97 = ?

    2    -3
102 x 97 = ?
102 -  3 = 99
97 + 2 = 99 (same answer)
2 x (-3) = (-06)   (base is 100, 2 zeros means 2 digit for answer.)
102 x 97 = 99(-06)
102 x 97 = 9,900 - 06
102 x 97 = 9,894

These are long when written, but just quick in the mind.

#Matamata

Completion by Addition and Subtraction

Adding 400 and 68 is as easy as

400 + 68 = 468 .

How about adding 598 and 78 ?

Add 2 to 598 to make it whole 600, making the addition simpler. Since 2 more is added, subtract 2 from the sum to get the final answer.

598 + 78 = ?

600 + 78 - 2 = 676

More examples:

44 + 19 = ?

44 + 20 - 1 = 63

97 + 86 = ?

100 + 86 - 3 =183

994 + 678 = ?

1000 + 678 - 6 = 1672

Same goes with subtraction. Subtracting 20 from 46 is as easy as

46 - 20 = 26 .

How about subtracting 18 from 54 ?

Add 2 to 18 to make it 20. Since 2 more is subtracted, add back 2 to get the final answer.

54 - 18 = ?

54 - 20 + 2 = 36

More examples:

444 - 198 = ?

444 - 200 + 2 = 246

55 - 29 = ?

55 - 30 + 1 = 26

875 - 496 = ?

875 - 500 + 4 = 379

#Matamata

Squares of Any Number

Case 1: Below a base number.

Base number can be 10, 100, 1000 or higher.

997^2 = ?

1000 - 997 = 3

997^2 = 997 - 3 | 3^2

997^2 = 994 | 009  ••

997^2 = 994009

•• Number of zeros of base number indicates number of digits for the last group of the answer. That is why 3^2 = 009 .

Other examples:

96^2 = ?

100 - 96 = 4

96^2 = 96 - 4 | 4^2

96^2 = 92 | 16

96^2 = 9216

9987^2 = ?

10000 - 9987 = 13

9987^2 = 9987 - 13 | 13^2

9987^2 = 9974 | 0169 ••

9987^2 = 99740169

Case 2: Above the base number.

13^2 = ?

13 - 10 = 3

13^2 = 13 + 3 | 3^2 •••

13^2 = 16 | 9

13^2 = 169

••• If a number is below the base, subtract. If a number is above the base, add. That is why 13 + 3 and not 13 - 3.

Other examples:

108^2 = ?

108 - 100 = 8

108^2 = 108 + 8 | 8^2

108^2 = 116 | 64

108^2 = 11664

1012^2 = ?

1012 - 1000 = 12

1012^2 = 1012 + 12 | 12^2

1012^2 = 1024 | 144

1012^2 = 1024144

Case 3: Near a multiple of base number.

204^2 = ?

Base number is 2*100. Follow the same procedure with case 1 and case 2, but this time, first group of the answer will be multiplied by base factor (2).

204^2 = ?

204 - 200 = 4

204^2 = 2*(204+4) | 4^2

204^2 = 416 | 16

204^2 = 41616

Other examples:

196^2 = ?

200 - 196 = 4

196^2 = 2*(196 - 4) | 4^2

196^2 = 2*192 | 16

196^2 = 384 | 16

196^2 = 38416

6080^2 = ?

6080 - 6000 = 80

6080^2 = 6*(6080+80) | 80^2

6080^2 = 6*6160 | 6400

6080^2 = 36960 + 6 | 400 ••••

6080^2 = 36966400

•••• Number of zeros of base number indicates number of digits for the last group of the answer. Carry over the excess digits to the first group ( 36960+6).

#Matamata