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Radicals – hand calculation of square root – pg.5/7

Posted in Math Theory by Lia on 06/01/2013

Radicals: pg. 1, 2, 3, 4, 5, 6, 7

 Hand calculation of square root

We start with a few basic things that will help understand the way this calculation is done.

A.  General notations used in square root calculation.

General notations used in square root calculation

B.  Grouping the radicand’s digits for an easier calculation.

To calculate by hand a square root we need to consider the radicand number first.

  • For the calculation we need to group the radicand’s digits (groups of two digits) on the right and on the left of the decimal point, as shown below:

Let’s assume that 1234.56” is the radicand.  

The grouping will be:

Radicand number's grouping_1_3

 

 

  • If the radicand’s integer  has an odd number of digits, the first group of digits, far left, will have only one digit as shown below.

Let’s assume that the radicand has the form “{}{12345.67}{}“.

The grouping will be:

Radicand number's grouping_2_4

 

 

  • If the radicand’s decimal area  has an odd number of digits,we add a “0” at the far right such, that the last group of digits will have two digits, as shown below.

Let’s assume that the radicand has the form “1234.5“.

The grouping will be:

Radicand number's grouping_3_5

 

 

  • In some cases, we add more groups of zeros to the radicand’s decimal to go on with the calculation ,to get a more  accurate result (for numbers that are not a perfect square), as shown below.

Let’s assume that the radicand has the form “1234“.

By adding zeros as decimals the grouping will be:

Radicand number's grouping_4_6

 

 

Examples of square roots with the radicand digits’ grouped:

Examples of square roots radicand digit grouping before doing the calculation_1

 

 

 

 

 

 

 

 

The calculation – Examples

 

a)  Steps for extracting a square root from a number that is an integer, a  perfect square and has an even number of digits. 

The example we use in this case is: “1369

 

The calculation Steps:

 

Step 1.  Place the number in the radicand area as shown bellow.

Example a, detail_1

Step 2.  Group the digits of the radicand.

Example a, detail_2_2

 

Step 3.  Try to find a single digit number that raised to the power two is smaller or equal with the first group of digit(s) on  the radicant’s left hand side.  In our case, the group of digits is 13 and the number that raised to the power two  smaller or equal with 13 is 3 ({}{3^2} = 9{}). Number 4 raised to the power two is 16, therefore  larger than 13.

 

Step 4. Place the number  found at Step 3 , in our case 3, multiply with itself, and the product,  in our case 9,  on the left hand side of the “Radical calculation divider line” as shown below.

Example a, detail_4_3

 

Step 5. The product obtain at Step 4, in our case 9, is placed under the first group of digit(s), in our case 13. Subtract 9 from 13 as shown.

Example a, detail_5_4

 

Step 6. The single digit number found at Step 3 , in our case 3, is placed in the result area, as shown.

 Example a, detail_6_5

Step 7. Add to the difference obtain at Step 5,  in our case 4, the  radicand’s  next group of digits, in our case 69 and we are going to have the number,  in our case 469needed to find the next result’s digit.

 Example a, detail_7_6

Step 8. To find the next result’s digit, start a new line on the left hand side of the “Radical calculation divider line”. It will start with a number that is two times the number in the result area, we have at this point (in our case 3). This number will be:  2 *3 = 6, as shown.

 Example a, detail_8_7

Step 9.  Find a single digit number “N“, such that the product between the number obtained by attaching “N” to the number found on Step 8  (in our case, number  6),  andN is lower or equal with the number obtain in Step 7 (in or case, number 469).

If “N” was equal with 7  would be  6N*N = 67*7 = 469. Means that “7 is the next result’s digit we are looking for. 

If “N” was equal with 8  would have been  6N*N = 68*8 = 544 >469″ title=”6N*N = 68*8 = 544 >469″/><img src=, therefore  “N cannot be  8

 Write 7 in the result area as shown.

Example a, detail_9_8

Step 10. The difference between the number obtain in Step 7 (number 469), and the number obtain at Step 9 (number 469) is 0. Because the radicand does not have any more digits and the last difference is 0, the calculation is over and the result of this radical extraction is the final number in the ” Radical’s result area”, as shown.

Example a, detail_10_9

 And the result is:

 

{}{sqrt{1369}} = 37{}

 

b)  Steps for extracting a square root from a number that is an integer, a  perfect square and has an odd number of digits. 

The example we use in this case is: “15376

The Steps to solve this radical are similar with the steps used to solve the radical in example a).

Use Steps 1 through 9 as used in example a) (the numbers may be different).

After Step 9, repeat Steps 7 through 9  until all the radicand’s digits are exhausted. At the end use Step 10 (as in example a))

Square root example_2_2

 And the result is:

 {}{sqrt{15376}} = 124{}

 

c)  Steps for extracting a square root from a decimal number with an even decimal number of digits, that is not a perfect square. 

The example we use in this case is: “2884.7691

The Steps to solve this radical are similar with the steps used to solve the radicals in examples a) and b).

When we arrive at the radicand’s decimal point, we place the decimal point in the result and continue our calculation.

Square root example_3_3

 And the result is:

{}{sqrt{2884.7691}} {approx} 53.71{}

the rest is: .0050

 

d)  Steps for extracting a square root from a decimal number with an even decimal number of digits, that is not a perfect square. 

The example we use in this case is: “36921.6942

The Steps to solve this radical are similar with the steps used to solve the radical in example c).

Square root example_4_3

 And the result is:

{}{sqrt{36921.6942}} {approx} 192.15{}

the rest is: .0717

 

e)  Steps for extracting a square root from a decimal number with an even decimal number of digits, that is not a  perfect square. 

The example we use in this case is: “19110.2982

The Steps to solve this radical are similar with the steps used to solve the radical in example d).

Square root example_5_4

 And the result is:

{}{sqrt{19110.2982}} {approx} 138.24{}

the rest is: .0016

 

f)  Steps for extracting a square root from a decimal number with an odd decimal number of digits, that is not a  perfect square. 

The example we use in this case is: “19110.298”

The Steps to solve this radical are similar with the steps used to solve the radical in example e).

Because the radicant’s two digit grouping for the last decimal group of digits, on the radicand’s far right hand, it is necessary to add an “0” as shown below.

Square root example_6_5

 And the result is:

{}{sqrt{19110.298}} {approx} 138.24{}

the rest is: .0011

 

g)  Steps for extracting a square root from a integer that is not a perfect square. 

The example we use in this case is: “24555

The Steps to solve this radical are similar with the steps used to solve the radical in previous examples.

To get a more accurate result, we add decimals,  groups of “00” as shown below.

Square root example_7_6

 And the result is:

{}{sqrt{24555}} {approx} 156.7{}

the rest is: .0011

 

Radicals: pg. 1, 2, 3, 4, 5, 67

 

 

 

 

 

 

 

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