The square root of ##89## is a number which when squared gives ##89##.
##sqrt(89) ~~ 9.434##
Since ##89## is prime ##sqrt(89)## cannot be simplified.
You can approximate it using a Newton Raphson method.
I like to reformulate it a little as follows:
Let ##n = 89## be the number you want the square root of.
Choose ##p_0 = 19## ##q_0 = 2## so that ##p_0/q_0## is a reasonable rational approximation. I chose these particular values since ##89## is about halfway between ##9^2 = 81## and ##10^2 = 100##.
Iterate using the formulas:
##p_(i+1) = p_i^2 + n q_i^2##
##q_(i+1) = 2 p_i q_i##
This will give a better rational approximation.
So:
##p_1 = p_0^2 + n q_0^2 = 19^2 + 89 * 2^2 = 361+356 = 717##
##q_1 = 2 p_0 q_0 = 2 * 19 * 2 = 76##
So if we stopped here we would get an approximation:
##sqrt(89) ~~ 717/76 ~~ 9.434##
Let’s go one more step:
##p_2 = p_1^2 + n q_1^2 = 717^2 + 89 * 76^2 = 514089 + 514064 = 1028153##
##q_2 = 2 p_1 q_1 = 2 * 717 * 76 = 108984##
So we get an approximation:
##sqrt(89) ~~ 1028153/108984 ~~ 9.43398113##
This Newton Raphson method converges fast.
##color(white)()##
Actually a rather good simple approximation for ##sqrt(89)## is ##500/53## since ##500^2 = 250000## and ##89 * 53^2 = 250001##
##sqrt(89) ~~ 500/53 ~~ 9.43396##
If we apply one iteration step to this we get a better approximation:
##sqrt(89) ~~ 500001 / 53000 ~~ 9.4339811321##
##color(white)()##Footnote
All square roots of positive integers have repeating continued fraction expansions which you can also use to give rational approximations.
However in the case of ##sqrt(89)## the continued fraction expansion is a little messy so not so nice to work with:
##sqrt(89) = [9; bar(2 3 3 2 18)] = 9+1/(2+1/(3+1/(3+1/(2+1/(18+1/(2+1/(3+…)))))))##
The approximation ##500/53## above is ##[9; 2 3 3 2]##