compare an integer number without using calculator

Question

we have $\displaystyle a=2^{12}$ and $\displaystyle b=3^8$ and I wonder if exist method to compare them without using calculator.

Really you need to show more effort than this and explain what you have tried. You will learn very little if you didn't spend some time puzzling it out for yourself. — Mark Bennet, Feb 28 '14 at 16:58

score 2 · Accepted Answer · answered Feb 28 '14 at 16:43

2

Take the fourth root of both, they are $2^3=8$ and $3^2=9$.

answered Feb 28 '14 at 16:43

alex

795

score 0 · Answer 2 · edited Feb 28 '14 at 16:43

0

Consider $\left(\dfrac{3}{2}\right)^8 = \left(\dfrac{9}{4}\right)^4 > 2^4 = 16 \Longrightarrow 3^8 > 16*2^8 = 2^4*2^8 = 2^{12}$. So $2^{12} < 3^8$.

edited Feb 28 '14 at 16:43

John Habert

4,001

answered Feb 28 '14 at 16:42

DeepSea

77,651

score 0 · Answer 3 · answered Feb 28 '14 at 16:42

0

Hint: if $p, q$ are positive integers with $p\lt q$ then $p^2\lt q^2$

answered Feb 28 '14 at 16:42

Mark Bennet

100,194

score 0 · Answer 4 · answered Feb 28 '14 at 16:57

You could, of course, do the calculations by hand, especially if you remember that $2^{10}=1024$, so that $2^{12}=1024\times4=4096$, in which case it suffices to make some crude estimates on $3^8$:

$$3^4=81\Longrightarrow3^6=81\times9\gt700\Longrightarrow3^8\gt700\times9=6300$$

or

$$3^4\gt80\Longrightarrow3^8\gt80^2=6400$$

compare an integer number without using calculator

4 Answers4