what is the easiest way to do squares by pencil and paper?

Question

math help? say it as you were trying to make a person from a different country understand

square roots

example: 175 *63=
the answer is 105
but how to i get to it?

Answer 1

It is very easy to come up with an estimate for the two whole numbers the square root is between. Take, for example, the sqrt of 5.

We know that 4 < 5 < 9. We choose 4 and 9 because they are perfect squares.

sqrt4 < sqrt5 < sqrt9
2 < sqrt5 < 3
So the square root of 5 is between 2 and three.


Following are two methods for finding the answer much more accurately.


1. Estimate the square root to at least 1 digit.
2. Divide this estimate into the number whose square root you want to find.
3. Find the average of the quotient and the divisor. The result becomes the new estimate.

The beauty of this method is that the accuracy of the estimate grows extremely rapidly. Each cycle will essentially double the number of correct digits. From a 1-digit starting point you can get a 4-digit result in two cycles. If you know a square root already to a few digits, such as sqrt(2)=1.414, a single cycle of divide and average will give you double the digits (eight, in this case).

In addition to giving a way to find square roots by hand, this method can be used if all you have is a cheap 4-function calculator. If students can get square roots manually, they will not find square roots to be so mysterious. Also, this method is a good first example of an itterative solution of a problem.



This other way is called Babylonian method of guess and divide, and it truly is faster. It is also the same as you would get applying Newton's method. See for example finding the square root of 20 using 10 as the initial guess:

GuessDivideFind average
1020/10 = 2average 10 and 2 to give new guess of 6
6 20/6 = 3.333average 3.333 and 6 gives 4.6666
4.666 20/4.666= 4.1414average 4.666,4.1414= 4.4048
4.4048 20/4.4048=4.5454 average = 4.4700
4.4700 20/4.4700=4.4742 average = 4.4721
4.4721 20/4.4721=4.47217 average = 4.47214
This is already to 4 decimal places
4.47214 20/4.47214=4.472132 average =4.472135
4.472135 20/4.472135=4.472137 average = 4.42136

Answer 2

Factorise it
175 * 63

=25 * 7 * 7* 9
=5 * 5 * 7 * 7 * 3 * 3Now taking the square roots for each number occuring twice u get number once in root

Root = 5 * 7 * 3 = 105

This method is useful mainly if the number is a perfect square.

However it can be used for non perfect numbers as follows.

27 =3 * 9 =3*3*3
Tking root
sqroot(27) = 3 *1.732

Answer 3

there's this technique our prof taught us, it's a long process but effective anyway:

you have to divide the number by squares

example., 2209

_ 4____
/ 2209
_16____ --> square of 4 is 16, then subtract like the usual
609 division then bring down the rest

then multiply 4 by 20 which is 80
then divide 609 by 80 and you'd get 7 point something
to see if you'd use 7 or 8
add 80 and 7 you get 87 then multiply 87 by 7 and you'd get 609
just attach 7 to 4 and you get 47,

so the square root of 2209 is 47., done

let's try your example,

175 * 63 = 11025

_10___
/ 11025
_100___
1025
--> 10*20 = 200
1025 / 200 = 5.125
200 + 5 = 205 * 5 = 1025

attaching 5 to 10 we get 105

so the square root of 11025 is 105., done

i know you could use this process with numbers that aren't perfect squares but i forgot how, maybe you'd just continue the process, not sure though.

hope you get the hang of it.,
it's pretty nice actually, hope it helps., =)

Answer 4

Write a table with 1 to 10 on the top increasing from left to right and 1 to 10 on the left, increasing as it goes down the paper.

Then fill in the product of the rows and columns. The squares will lie on the diagonal going down the paper from upper left to bottom right.

Answer 5

MULTIPLY THE NO. BY NEAREST MULTIPLE OF 10 AND MULTIPLY THE REMAINING BY NO. AND ADD BOTH THE RESULTS.FOR EG- SQ OF 59
59*50=2950
59*9=531
R=3481
SECOND IS BY USING THE FORMULA(a+b)*(a+b)
(50+9)(50+9)=2500+81+2*50*9=3481

Answer 6

Drat. I thought you meant the game. For that, just use grid paper.
Now I suppose I should look it up.

Or YOU could... This IS the WWW after all! Search "Square Root Formula".

Answer 7

The same way you do multiplication, since squareof a number is that number multiplied by itself.

Answer 8

drawing them use graph paper or a ruler or a stencil

Answer 9

u divide the no into its prime factors and rearrange them to get two same nos

Answer 10

Here you go!
http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html