They are advanced because unlike ordinary multiplication and division, their operations (methods of getting the answers), are indeed, much more tricky and difficult.
SQUARE OF A NUMBER
Getting the ‘square value’ of a number is like doing a special kind of multiplying a number, in which both the multiplicand and the multiplier are equally the same values. Sometimes, it is described as product of a number multiplied by itself.
Examples:
2 x 2 = 4
27 x 27 = 729
146 x 146 = 21,316
The value 4 is sometimes called the ‘square value’ of 2, or simply, “square of 2”. The same way, 729 is the “square of 27” and 21,316, the “square of 146”. Sometimes, instead of writing 2x2, 27x27 or 146x146, “a small number 2 in upper right side” of a given number is used as a symbol, telling you to multiply that number by itself. So, instead of 2x2, we write 22 = 4 and 27x27 as 272 = 729, while 146x146 as 1462 = 21,316.
Maybe, you are wondering why it is called ‘square’. Probably, early mathematicians noticed that the measure of the area of a square is always equal to a certain ‘number multiplied by itself’, so they named it, that way.
SQUARE ROOT
On the other hand, getting the square root of a number, needs a very different way, of dividing a number. Unlike in ordinary division at which you need to mention the value of the divisor, in getting the square root of a number, both the divisor and the quotient are unknown and the difficult thing is, both divisor and the quotient must be equally in the same values.
Examples:
36 ÷ 3 = 12 36 ÷ 4 = 9 36 ÷ 6 = 6
In the above examples, 36 can be divided by 3 or 4 but the quotient would not be equal or the same with the divisor. Dividing 36 by 6, we can get a quotient equal to 6, which is the same exact value as to the divisor. In this situation, we can say then, that, 6 is a square root of 36.
In doing this special kind of division, the symbol √ is used before a given number (example √144, read as, “the square root of one hundred forty-four’), to tell you to look for a divisor that will give a quotient, equal to that divisor. Dividing 144 by 12, we come up with a quotient equal to 12 (144 ÷ 12 = 12). Showing equal values for both divisor and quotient we can say then, that √144 = 12.
But there are occasions that the given numbers are in large values (example, √139,876). Getting the square root of such large valued numbers requires a very tedious and tricky method called ‘long hand division’. But as a practice, small valued numbers are introduced for grade school children, to make them easier to memorize.
TABLE OF SQUARE ROOTS
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√81 = 9
√100 = 10
PERFECT SQUARES
Not all numbers from 1 to 100 give a square root in whole exact values. Most of them are in decimal values. Below is a list of examples of numbers, having no whole exact square root values:
√ 2, √3, √10, √99 , √28 , √50
Counting from 1 to 100, there are only ten numbers having square roots in ‘exact whole values’ and they are called perfect squares (or simply call them “PERKS”).
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
(Using a calculator, find the square roots of each numbers from 1 to 100 and write down which numbers have an exact whole numbers)
SET OF “PERFECT SQUARES” = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
For fun way of naming things, let's call them Perkies, pertaining to "squares of whole numbers".
TABLE OF SQUARES
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
