What is the square root of 3.4?

1 Answer
Sep 18, 2015

Answer:

#sqrt(3.4) = sqrt(85)/5 ~~ 1.8439#

Explanation:

If #a, b >= 0# then #sqrt(a/b) = sqrt(a)/sqrt(b)#

If #a >= 0# then #sqrt(a^2) = a#

So:

#sqrt(3.4) = sqrt(85/25) = sqrt(85)/sqrt(25) = sqrt(85)/5#

Here I have chosen to express #3.4# as #85/25# to get the smallest whole value for the denominator.

I could also have written #sqrt(3.4) = sqrt(17/5) = sqrt(17)/sqrt(5)#

If I wanted to calculate an approximation for #sqrt(3.4)# by hand then I would probably have used this instead:

#sqrt(3.4) = sqrt(340/100) = sqrt(340)/10#

Then I would work out an approximation for #sqrt(340)# and divide by #10#.

For example: #18^2 = 324# and #19^2 = 361#, so:

#18 = sqrt(324) < sqrt(340) < sqrt(361) = 19#

Using a Newton Raphson type method to find #sqrt(340)#, I might chose #18.5 = 37/2# as my first approximation #a_0#, then make better approximations using the formula:

#a_(i+1) = (a_i^2 + n)/(2a_i)#

where #n = 340#.

Then

#a_1 = (37^2 + 340 * 2^2) / (2 * 37 * 2) = (1369+1360)/148#

#= 2729 / 148 ~~ 18.439#

So #sqrt(340) ~~ 18.439# and #sqrt(3.4) ~~ 1.8439#