Between what two consecutive integers do #root3(16)# lie?

Question

3600 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-10-13T18:54:55

#2 and 3#

Pressing #root(3)16# into a calculator gives #2,5198#

This decimal value hence lies between the integers #2# and #3#.

That is, #2,3 in ZZ and root(3)16 in [2;3]#

Answer 2 · 2015-10-13T19:13:30

Between #2# and #3#

#2^3 = 8 < 16 < 27 = 3^3#

and #f(x) = x^3# is strictly monotonic increasing, so

#2 = root(3)(2^3) = root(3)(8) < root(3)(16) < root(3)(27) = root(3)(3^3) = 3#