Question

what is the cube root of 3456

Answer 1

You can always guess.

14*14*14 = 2744
15*15*15 = 3375
16*16*16 = 4096

Well, that was pretty good guessing.

15*15*15 = 3375 <== From Above
15.1^3 = 3442.951
15.2^3 = 3511.808

15.1^3 = 3442.951 <== From Above
15.11^3 = 3449.795831
15.12^3 = 3456.649728 <== Just a hair over the integer portion.
15.119^3 = 3455.963930159
15.1191^3 = 3456.032505860871
15.11905^3 = 3455.998217896542625
15.119051^3 = 3455.998903653606689651
15.119052^3 = 3455.999589410761468608 <== 3 9's is pretty good.
15.1190525^3 = 3455.999932289372875953125 <== 4 9's!
15.1190526^3 = 3456.000000865097878851576 <== 5 0's! (Didn't count the last 0 because of rounding.)

Of course, using the \(\left[x^{y}\right]\) button on your calculator might be faster.

3 4 5 6 \(\left[x^{y}\right]\) ( 1 / 3 ) = 15.119052598738477977206527287339

Wasn't that fun?