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

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

## Answers and Replies

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
Arithmetic algorithms can approximate these square roots, but because they are all irrational, the decimal representations are non-repeating and non-terminating.

HallsofIvy
Science Advisor
Homework Helper
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
Mentor
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
Science Advisor
Gold Member
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.

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.