Did I mention the same method may be used to find cube roots? Cube root of six.
= 1.8 x cbrt 6/5.832
= 1.8 x cbrt 1.0288..
cbrt of 1 plus a small number slightly exceeds 1 plus 1/3 of that number. So we take 1.027, which gives us 1.009 as the close cube root we seek.
1.8 x 1.009 cbrt 1.0288../1.02724..
1.8 x 1.008 = 1.8162, which cubed, gives us 5.9988..
If we take it another step, we have
1.8 x 1.009 x sqrt 1.0015, which is approximately 1.8 x 1.009 x 1.0005 = 1.817.., which cubed, yields 5.99987..
Fourth root may be arrived at by the same method, remembering that the fourth root of 1 + a small number = approximately 1 plus 1/4 that number.
For working engineers, it sometimes comes in handy to be able to calculate a square or cube root manually, particularly if your calculator has run out of batteries or if you just happen not to have one with you.
Of course, it is important for students to understand that they can get a close approximation to a square root, how it is done. That goes without saying, although a number of those who have commented here tend to doubt this.
I believe that students should have the knowledge to make their own tables, or to pretty much, in principle, to calculate by hand what the calculator does for them.