6 digit square root 377300

Question

How do i go about finding the square root of 377300?

I've tried some video tutorials however, i can't get it to work.

i pair the numbers 2 by 2 by 2

37 73 00

closes for 37 is: 6 square = 36

the square root of 00 is: 0 square = 0

so remainder of 37-36 = 1 then i bring down the 73 = 173

6x2 = 12_ < supposedly multiplying the number i choose to put would give me 173

for example 121x1, 122x2, 123x3.

but nothing gives me 173

1 gives me 121

so 173-121= 52

so 37 = 6, 173 = 1, 00 = 0

6+1+0 = 7x7 = 49

Anyway, i'm probably missing a big step. Teach me! i know that 4900x77 = 377300 or 70/77

Answer 1

I almost never look at the answer supplied by another tutor but I was curious to see how Lauren H. responded. Her method is quite correct, i.e factoring the number and extracting square roots of the factors. Of course, you still have to extract the square root of 7 and 11! Unfortunately I think Lauren made a typo in her last line which should read 70*sqrt(7)*sqrt(11).

However, what David D. is trying to do is to extract the square root manually and he is fouled up because the method is not clear to him.

David, you are starting off correctly, by setting of the number into 3 pairs of digits. The largest square in 37 is 6. Square 6 and subtract it from 37 which leaves 1. Bring down the next group so that you have 173 .

Now double 6 and divide 12 into 17 which is 1. Append the 1 to the 12 and put the one above the square root line. Multiply the 1 by the 121 and subtract the 121 from 173 to 52. Bring down the 2 zeros.

Double the 61 to get 122 and divide it into 5200 to get 4, Put the 4 above the square root line and append it to 122 to get 1224. Multiply 1224 by 4 to get 4896. Now keep on going in this fashion (bringing down 00) until you get the desired number of decimal places in the answer. This is a cumbersome process, but ultimately gives the exact result. Please note that this method is based on the square of a binomial , i.e.

(a + b)² = a² + 2ab + b².

The other method is Newton's method which (as far as I know) is still the method used by the calculator and the computer. It is iterative and here is the iterative formula:

x_n+1 = (1/2)[x_n + (N/x_n)] where N is the of which you want the square root.

In this present case you would start with 600 (or any better first approximation). and get a successively better approximation. For example if x₁ = 600, x₂ = 613.4. The next iteration will be correct to 4 significant figures and the following iteration will be correct to 8 significant figures...but it is also cumbersome.

I apologize both to you and Lauren H. for this long answer, but I thought it might be useful for you to have some background for the simple question you asked.

Answer 2

the answer is 614.2475071

I used a calculator and the sq rt function key.

 4900x77 = 377300 This is true.

So, that means that 100 x 49 x 77 = 377300

So, √100 = 10, and √49 = 7

7 x 11 = 77

so √377300 = 10 x 7 x √7 x√11 = 17√11√7