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Square a number

The product of a number times itself is the square of that number. So 3 squared is 3 times 3 or 9.

The calculator has a squared key. It is quite useful to minimize data entry for difficult numbers.

Square Root of a number

The square root of a number is that value which when multiplied by itself produces the number, itself. So the square root of 9 would be what number when multiplied by itself would yield 9? Well in your head you can figure it to be 3. 3 times 3 equals 9. So the square root of 9 is 3. 

The calculator has a square root key which makes the process painless to find the square root of a number.

Note: When there are two factors under the radical (square root sign) then you must first perform what ever math operation is identified for the factors, like division or multiplication. Then press the equals sign. This will yield the result of what is to be done with the factors.

Lastly, press the square root key to find the square root of the quantity under consideration.

 

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Extras For Experts

In the department of more than you ever wanted to know, here is the technique of solution to find the square root of a number without a calculator, as if anyone would ever want to do that.

square root

(algorithm)

Definition: This describes a "long hand" or manual method of calculating or extracting square roots. Calculation of a square root by hand is a little like long-hand division.

Suppose you need to find the square root of 66564. Set up a "division" with the number under the radical. Mark off pairs of digits, starting from the decimal point. (Here the decimal point is a period (.) and a comma (,) marks pairs of digits.)

               ___________
             \/  6,65,64.
 

Look at the leftmost digit(s) (6 in this case). What is the largest number whose square is less than or equal to it? It is 2, whose square is 4. Write 2 above, write the square below and subtract.

               __2________
             \/  6,65,64.
                -4
               ----
                 2
 

Now bring down the next two digits (65). The next "divisor" is double the number on top (2x2=4) and some other digit in the units position (4_).

               __2________
             \/  6,65,64.
                -4
                -----
             4_ ) 265
 

What is the largest number that we can put in the units and multiply times the divisor and still be less than or equal to what we have? (Algebraically, what is d such that d 4d ≤ 265?) It looks like 6 might work (since 6 * 40 = 240), but 6 is too big, since 6 * 46 = 276.

               __2__6_____
             \/  6,65,64.
                -4
                -----
             46 ) 265
                  276   TOO BIG
 

So try 5 instead.

               __2__5_____
             \/  6,65,64.
                -4
                -----
             45 ) 265
                 -225
                 -------
                   40
 

Repeat: bring down the next two digits, and double the number on top (2x25=50) to make a "divisor", with another unit.

               __2__5_____
             \/  6,65,64.
                -4
                -----
             45 ) 265
                 -225
                 -------
             50_ ) 4064
 

It looks like 8 would work. Let's see.

               __2__5__8__
             \/  6,65,64.
                -4
                -----
             45 ) 265
                 -225
                 -------
             508 ) 4064
                  -4064
                  ------
                      0
 

 

So the square root of 66564 is 258. You can continue for as many decimal places as you need: just bring down more pairs of zeros.

Why does this work?

Consider (10A + B) = 100A + 2 10AB + B and think about finding the area of a square. Remember that 10A + B is just the numeral with B in the units place and A in the higher position. For 42, A=4 and B=2, so 10 4 + 2 = 42.

diagram of a square that is 10A + B on a side showing a 100A squared  rectangle, two 10AB rectangles, and a B squared rectangle

The area of the two skinny rectangles is 2 10A B. The tiny square is B. If we know A and the area of the square, S, what B should we choose?

We previously subtracted A from S. To scale to 100A, we bring down two more digits (a factor of 100) of the size of S. We write down twice A (2A), but shifted one place to leave room for B (10 2A or 2 10A). Now we add B to get 2 10A + B. Multiplying by B gives us 2 10AB + B. When we subtract that from the remainder (remember we already subtracted A), we have subtracted exactly (10A + B). That is, we have improved our knowledge of the square root by one digit, B.

We take whatever remains, scale again by 100, by bringing down two more digits, and repeat the process.

 

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