How can I do square roots in my head?

Answer 1

Hi:

there a five ways to do Square roots in your head

If you are very good at addition,multiplying and division then use the Newton -Raphton method:

Gn= (SQRN/Gn+ Gn)/2

repeat using the new Gn each time for about ten times

Gn = guess number
SQRN = Number to square rooted

for example:

Square root of 26
Run # 1

Gn= 5

Gn= (26/5+ 5)/2

Gn= (5.2+5)/2

Gn= 10.2/2

Gn= 5.1

End of run #1

Run # 2

Gn = 5.1

Gn= (26/5.1+ 5.1)/2

Gn= (5.098039+5.1)/2

Gn= (10.198039)/2

Gn= 5.0990196

End of run #2

Do this for ten times and the Square root of 26
which appox equals 5.0990195


another way is to factor the Number being Square root

if you know the basic 1 to 12 roots

For example the square root of 8

8= 2*2*2

So (2*2)*2 = 4*2

and four square rooted is 2 So

Sqr(8) = 2 * Sqr(2)

the Square root of 2 = 1.14142135

So Sqr(8) appox = 2.82842713

#3 the School boy way see www.nist.gov/dads/HTML/squareRoot.html

as to how to do that. Which is the best method to do them. If you can't do the above or below

# 4 ) the Bablyon method

Using S=(x+y)^2 ( y being the the closes square of the number without going over it see the example

S= x^2+2xy+y^2

S-y^2 = x^2 + 2xy
S-y^2 = x(x+2y)

S-y^2
------- = x(x+2y)/(x+2y)
(x+2y)

S-y^2
------ = x
(x+2y)

For example the Square root of 5

5= (x+2)^2 = (x+2)(x+2)

5= (x^2 +4x+4) ( y=2 here so 2*2 = 4)

5-4=x^2+4x+4-4

1= x^2+4x

1= x(x+4)

1/(x+4)=x(x+4)/(x+4)

1/(x+4) = x

Run 1

1/(1+4)= x

1/5= x

x=.2

run 2

1/(.2+4)= x

1/4.2= x

x= 0.2359550

run 3

1/(0.2359550+4) = x

x= 1/ 4.2359550

x = 0.235955

run 4

1/(0.235955+4) = x

1/(4.2359550) = x

0.236074 =x

run 5

1/(.236074+4) = x

1/(4.236074) =x


4.236074= x

0.2360676 =x

End of run 5

(2+x)^2 = 5

x = 0.2360676

So 2.2360676 is the square root of five

2.2360676 ^2 = 4.99999 close enough for government work

# 5 memerize the logarithm table one thru 1000

and squares and square roots and cube cube roots and any other powers or roots is a breeze

Hope this helps

Answer 2

You can learn the square roots of relatively square numbers the same way you'd learn a multiplication table, that is, memorization. For example, the square root of 144 is 12, because 12*12 = 144.

To take the square roots of numbers that are not square, you can't really do that in your head. However, you can do it on paper with a reasonable degree of accuracy, using a formula based on derivatives. The square root of a^2 + x is very close to a + x/2a. For example, 102 is 10^2 + 2. 10 + 2/20 = 10.1. The actual square root of 102 is 10.0995, a difference of only 0.0005.

I'll write out the general formula, but it won't make much sense if you don't already know derivatives. Basically, f(x + d) ~= f(x) + d*f'(x). The smaller d is, the more accurate the approximation. The derivative of sqrt(x) is 1 / (2*sqrt(x)).

Answer 3

just remember the squares of all the numbers.
or divide the number in two parts. write it in addition form. then use thar formula.i.e a^2+b^2+2ab. where a and b are the two parts of the numbers. and if u find an easy way from this. please kindly inform me too. my id is [e-mail]

Answer 4

I have them memorized until 144 very easily. I'm sure if you put some time into it you could go well beyond 400 but other then that I'm not sure.

Answer 5

Practice ☺


Doug

Answer 6

some but many have to be worked out on paper