Which of the points on the number line above best expresses the expres

is there any other way to get this ??

Hi, apologies. I will be clearer from now on. I tried to use approximations based on powers and cyclicity. So 2^1 = ends in 2, 2^2 = 4, 2^3 = 8 and similiarly, for 4: 4^1 = 4, 4^2 = 16, 4^3 = 64. So the ending number matters. I don't know if this is the best solution, but it saved time for me,and seemed fine to apply as a theory

root 2(1-root2) = 1.414(1 - 1.414) = approximately with ending numbers 4*4 = 16 . So the number could be (negative) 0. (some number of zeroes) followed by 6. So I took -0.6 (as a possible approximation)

Root 3(1- root 3) = 1.73(1- 1.73) = 1.73 (-0.73) = approximately ending in 0...(some zeroes) 9 (3*3 = 9)

Then I combined the numerator/ denominator = 0.6/0.9 (it becomes positive now, as -ve cancels out) --> this is approximately 0.2/0.3 which 0.66 just a little more than half. B seemed suitable.

Please let me know if there are any errors conceptually or otherwise. Would love to know it, and hopefully not repeat the same error. Thank you!

Madhavi1990 , I like this solution, but I don't understand
1) how you got TO -.414, e.g.

You did not multiply by -1. That would have yielded \(\frac{2-\sqrt{2}}{3-\sqrt{3}}\approx{\frac{.59}{1.27}}\);

And 2) how you got from \(\frac{-.414}{-.713}\) to \(\frac{0.6}{0.9}\).

Maybe I'm slow on the uptake today. I don't understand your shorthand here, for example :


Would you please explain how you derived the first fraction, then how you got from the first fraction to the second? (And in steps with full words?)