Algebra - How do I find the square root of numbers without a calculator?

Question

Here are the ones I need help with:
Square root of 420
Square root of negative 5.80
Square root of +/- 1158

Answer 1

a quick estimate is to figure the numbers it falls between... we know 20 squared is 400 and 21 squared is 441. 420 is roughly midway between 400 and 440, so my initial guess would be a number approx. midway between 20 and 21. assume it's 20.5 and verify. 20.5 squared is 420.25... not bad eh? the same rule should apply for your following two numbers. for negative numbers you use what are called imaginary numbers... solving works the same except you end the number with the letter i. i.e. the square root of negative 4 is 2i.

Answer 2

1) 420

(420)^.5 = (3 x 4 x 5 x 7)^.5 = (4)^.5 x (3 x 5 x 7)^.5
= 2 x (105)^.5 = approximately 2 x 10 = 20

2) -5.80 geez.

(580 / 100)^.5 x (-1)^.5 = ((4 x 5 x 29)^.5 / 10) x i = 2x (145)^.5 / 10 i

= approximately 2 x 12 /10 i = 2.4 i
(since 144 = 12 ^2)

3) +/- 1158 same way. break into low numbers and square root when possible

(1158)^.5 = (2 x 3 x 193) ^.5 = approx = (2 x 3 x 2 x 100)^.5
= (4 x 100 x 3)^.5 = 2 x 10 x (3)^.5 = approx = 2 x 10 x 1.7 = 34

for -1158 it would be 34i

Answer 3

Unless the numbers are perfect squares, it will be tough to do it without a calculator.
1) sq rt 420 = 20.4939
2) sq rt -5.80 = not possible. A number squared cannot be a negative number. Since the definition of any number squared is the number multiplied by itself, the answer is always positive.
3) sq rt 1158 = 34.0294

Answer 4

If you have a table of logarithms find the log of the number, divide by 2 and look up the antilog.
There is a manual way which is a modified form of long division but I doubt that I can explain it so you could understand it here. the answers to the above are:

20.494
2.408i
34.029 and 34.029i
the square root of a negative number is imaginary, that is why the i is shown on two answers

Answer 5

You are breaking basic rules of computing!

There are 2 ways to compute!

1) Find solution to 'any typical computing problem'.

2) Compute 'in a known manner' relating a known range of numbers!

a) Your problem is under catagory (1) as given above!
b)You have fixed which numbers you want to solve and (c) calculater should not to be used! So your solution to problem involve 3 complexities! It is not possible to do by most human -minds!

However I can give you some tips to find square root of some type of numbers in a known zone!

50, 100 and numbers near to 50, 100 have squares that are as given below!

45^2= 2025 ..... 95^2=9025
46^2= 2116...... 96^2=9216
47^2= 2209 ...... 97^2= 9409
48^2= 2304....... 98^2= 9604
49^2= 2401...... 99^2= 9801

50^2= 2500...... 100^2= 10000

51^2= 2601...... 101^2= 10201
52^2= 2704...... 102^2= 10404
53^2= 2809...... 103^2= 10609
54^2= 2916...... 104^2= 10816
55^2= 3025...... 105^2= 11025
Similarly,
500 and 1000
5000 and 10000
50000 and 100000
500000 and 1000000
------------ and ----------- also follow same principle!

Catagory (2) computing principle is explained above , which anybody can learn according to ability of a user!

Ability means --> 500001^2= 250001000001 (mentally)

If you devide it by 9 we get 2777788889. it is (166667)^2

You may use said method to find square root also!


49^2= 2401..... (25 -01) 01^2
51^2= 2601 ....(25+01) 01^2
52^2= 2704 ....(25+02) 02^2

means you either add or subtact difference of number to be squared from 50 and then add square of difference!

For numbers near hundread difference is doubled but adding square is similar!

No magic will compute but logic will do it, which give a trillion digit answer also!

Answer 6

Sq. rt. of 420 = -/ LCM of 420

= -/ 2 x 2 x 3 x 7 x 5

= 2-/ 3 x 7 x 5

= 2 (3)^1/2 (7)^1/2 (5)^1/2

= Antilog (log 2 + 1/2 log 3 + 1/2 log 7 + 1/2 log 5)

= Antilog [1 + 1/2 log (3 + 7 + 5)]

= Antilog (1 + 1/2 log 15)

I think this will help you!

Answer 7

to find the sq root u divide
the no. into pairs starting from the right side
eg: _ __
169--> 1 69
now:
then u start from the left and dive
the leftmost pair by its nearest root
then 2x the no. and divide this no.
by the next no. generated
its sort of like this:see eg. below
when you double the no. u join it w/ another
no. that when the new no. formed
multiplied w/ the joined no. gives you the no. pair inside
confusing? i know. hope this helps:
..._13______
1/ 1 69
.....1
------------
23./.......69
..............69
-------------------
xxxxxxxxxxxxxx
therefore the root=13

Answer 8

Yes.

Look up your wuestions in yahoo questions
it has been asked and answered before..

http://answers.yahoo.com/search/search_result;_ylt=Atbiaz6.HKkQCpZH8GEdAs8azKIX?p=square+root+of+numbers+without+a+calculator

Answer 9

Try these websites to help you out with the square root problems:
http://en.wikipedia.org/wiki/Square_root
http://math.arizona.edu/~kerl/doc/square-root.html
http://www.geocities.com/oosterwal/puzzle/squareroots.html
http://mathforum.org/library/drmath/view/51890.html

I hope these websites help you out with your math problems.