How do you calculate #4sqrt(5)# ?

3 Answers
Oct 7, 2017

See below.

Explanation:

#4sqrt(5)# is also expressed as #4# times #sqrt(5)#, or #4*sqrt(5)#.

If you were to type #sqrt(5)# in your calculator, you would get a value of #2.2360679775#. Then, take this and multiply it by #4# to get the final answer, #8.94427191#.

I hope that helps!

Oct 7, 2017

#8.9442#

Explanation:

#4sqrt5=4*2.2361=8.9442#

Oct 7, 2017

#4sqrt(5) ~~ 51841/5796 ~~ 8.94427191#

Explanation:

Note that #(4sqrt(5))^2 = 4^2*5 = 16*5 = 80 < 81 = 9^2#

So #4sqrt(5)# is an irrational number a little less than #9#.

Note that in general:

#sqrt(a^2+b) = a+b/(2a+b/(2a+b/(2a+b/(2a+...))))#

So we can put #a=9# and #b=-1# to find:

#4sqrt(5) = sqrt(80) = sqrt(9^2-1) = 9-1/(18-1/(18-1/(18-1/(18-...))))#

We can terminate this generalised continued fraction early to get rational approximations to #4sqrt(5)#.

For example:

#4sqrt(5) ~~ 9-1/18 = 8.9bar(4)#

#4sqrt(5) ~~ 9-1/(18-1/18) = 9-18/323 = 2889/323 ~~ 8.944272#

#4sqrt(5) ~~ 9-1/(18-1/(18-1/18)) = 9-1/(18-18/323) = 9-323/5796 = 51841/5796 ~~ 8.94427191#