On Finding Square Roots

Multiplication is one of the four basic operations in arithmetic. It is repeated addition. For instance,

5+5+5=15 can be written in multiplication as 5x3=15 or (5)(3)=15

But what about repeated multiplication? We call it exponentiation. As an example,

8x8x8x8=4,096 can be expressed in exponential form as 8⁴=4,096

The case of exponentiation where a number is used twice as a factor (or multiplied by itself) is called squaring the number, and the resulting product is called the square of that number. For example,

5²=25 because 5x5=25

The inverse process, where we instead find a number that when multiplied by itself will give us the number in question, is called finding the square root. In the example just mentioned,

If 5²=25, then 5 is said to be the square root of 25

Before we describe the procedure of finding square roots, let's have first a view of some squares of numbers.

Number, x	Square, x2	Number, x	Square, x2
1	1	6	36
2	4	7	49
3	9	8	64
4	16	9	81
5	25	10	100

                             Table 1. Squares of 1-digit numbers

Number, x	Square, x2	Number, x	Square, x2
11	121	69	4761
22	484	78	6084
34	1156	87	7569
43	1849	96	9216
55	3025	99	9801

                             Table 2. Squares of 2-digit numbers

Number, x	Square, x2	Number, x	Square, x2
101	10201	666	443556
209	43681	782	611524
354	125316	875	765625
473	223729	960	921600
567	321489	998	996004

                             Table 3. Squares of 3-digit numbers

The Meaning of Square Roots

If y² = x, then y is called the square root of x.

To find the square root of a number, we find the number which when multiplied by itself will give you the given number.

From the given tables, we note the following:

1.       The square root of a one-digit number is a number having one or two digits.

2.       The square root of a two-digit number is a number having three or four digits.

3.       The square root of a three-digit number is a number having five or six digits.

Then if a number whose square root we wish to find is separated into groups of two digits each, starting at the decimal point, the number of two-digit groups would be the same as the number of digits in the square root. For instance,

388,129 contains three groups (38’81’29),and its square root is 623, a three-digit number.

Here are the steps in finding the square root of a number:

1.       Group the digits two by two, starting from the decimal point.

2.       Determine the largest square less than or equal to the first group. Write it below the group and write its square root above the group.

3.       Subtract, then bring down the next group.

4.       Multiply the partial answer by 20.

5.       By trial and error, find a number from 1 to 9 that when added and multiplied to the result in step 4, will approach or equal the result in step 3.

6.       Write the trial number in step 5 above the next group, then write the product below the result in step 3.

      7.   Repeat steps 3 through 6 until all groups have been brought down.

The following example will illustrate these steps:

Find the square root of 527,076.