sqrt of 101 . please full solution

Question

please send me full solution

thanks

Answer 1

Square root of 101 is about 10.0499. But there are two square roots: +10.0499 and -10.0499. A good approximation to sqrt(101) is 10(1+(1/2)*(1/100))=10(1 1/200)=10 1/20 (ten and one twentieth).

Here's a neat way of working out the square root of 101. We know the square root of 100 is 10, right? So the square root of 101 will be 10 plus a little bit we'll call x: (10+x)^2=101. Expand the brackets: 100+20x+x^2=101, which simplifies to x^2+20x=1. Here's the interesting bit: x(x+20)=1, so x=1/(x+20). We now use an iterative process to find x quickly. We start the process by putting x=0 on the right-hand side; that gives us a new value of x=1/20 or 0.05. If we add 10 to this we have 10.05 (201/20) and this is our first estimate of the square root of 101. Let's see how close it is by squaring 10.05: 101.0025. Pretty close! But we can do better. The new value of x, remember, was 1/20, so we put this into the expression on the right to get the next value of x: x=1/(401/20)=20/401. Add 10 and we get 4030/401. Square this and we get 100.9999938 or thereabouts. That's even closer. If we add 20 to 20/401 we get 8040/401. So our next value of x is 401/8040. Add this to 10 and we get 80801/8040=10.04987562, which is a very very good approximation to the square root of 101.

Answer 2

[Commentary added by Rod]

sqrt 101=

10.0498 [using square root "long division", which consists of working out a single digit for every pair of digits in the number we are trying to take the square root of. The number is split into pairs of digits working to the left and right of the position of the decimal point.]

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sqrt 101.00000000     

    - 100 [subtract nearest perfect square to 101, which is 10 squared: note the first two digits of the answer is 10]

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          100 : 200=0 [after subtracting 100, 1 is the remainder and we add two zeroes to the end to continue. We double the answer we have so far, that is 2*10=20 and we make another number out of it that looks like 20? where ? is the next digit, then, by comparing 100 with 20?, we work out what 20? times ? would come to so that it can be subtracted from 100. Since 20? cannot be subtracted from 100, ?=0, so the next digit in the answer is 0. Therefore we have 10.0 so far as the answer, because the decimal point aligns with the decimal point of the number whose square root we seek.]
          10000 :2004*4 [here we added another two zeroes to the end of our remainder making 10000. So far the answer is 200, ignoring the decimal point. Add ? to the end as the next digit and multiply it by ?: 200? times ?. Again we pick ? so that the product is as close to 10000 as we can get without exceeding it. Compare 10000 with 200? times ?. Try ?=4 so the product is 2004*4=8016, as shown below, so we subtract 8016 from 10000.]

        -  8016

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           198400 : 20089 * 9 [the remainder is 1984 so we add a pair of zeroes to make 198400. The answer so far is 1004, which, when doubled, is 2008. So we want 2008? times ? to be comparable with 198400. Try ?=9 and do the multiplication 20089*9=180801.]

        -  180801

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               1759900 : 200988 * 8 [the new remainder is 17599 to which we add another pair of zeroes and repeat the process for the next ?, which turns out to be 8. 

          -    1607904 [200988*8=1607904.]

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                151996 [this remainder suggests that the next ? will be 7, but so far we have 10.0498 as the square root. If we were to round it up because the next expected digit is 5 or more, the square root would be 10.0499.]