How Do You Find The Exact Value Of Square Root of 3, 5, 7, 11?

[mymachine]

Is there any method to find the exact value of the square root of 3,5,7,11,13,14,15,17,18, etc.?

Thank you

 

[SteamKing]

Arithmetic algorithms can approximate these square roots, but because they are all irrational, the decimal representations are non-repeating and non-terminating.

 

[HallsofIvy]

The exact values of the square root of 2, 3, 5 ,7, 11, etc are $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{7}$, $\sqrt{11}$. That's the best you can do. As SteamKing said, all of those, and, in fact, the square root of any integer that is not a "perfect square", are irrational- they cannot be written as a terminating decimal, they cannot be written as a repeating decimal like "0.14141414...", and cannot be written as a fraction (integer over integer).

(I added "2" to the beginning of your list. I am surprized you did not have it.)

 

[Mark44]

Might as well add 6, 8, 10, and so on to the list, since none of these is a perfect square, and consequently does not have a square root that is rational.

 

[lavinia]

If instead of a infinite decimal expansion you would accept some other infinite expression then you can express the square root of 2 as an infinite continued fraction.

 

[micromass]

If instead of a infinite decimal expansion you would accept some other infinite expression then you can express the square root of 2 as an infinite continued fraction.

The infinite fraction representation is a really nice one because it exhibits a lot of regularity. In the decimal expansion of $\sqrt{2}$, there is no way to know which decimal comes next. But the infinite fraction is very straightforward and exhibits a nice pattern.