Estimating powers with rational exponents

Question

The task is to predict the order of the following six expressions from lowest to highest, without the aid of a calculator:

1. $\sqrt[4]{623}$
2. $125^{\frac{2}{5}}$
3. $\sqrt[10]{10.24}$
4. $80.9^{\frac{1}{4}}$
5. $17.5^{\frac{5}{8}}$
6. $21.4^{\frac{3}{2}}$

Estimating expressions 1 and 4 are easy because the fourth root of 625 is 5, and the fourth root of 81 is 3. But for the other ones I'm really not sure how one would go about it.

For expression 2, I suppose you could reason that $2^5=32$ and $3^5=243$, so the fifth root of $125$ is roughly $2.5$, and $2.5^2=6.25$ (which isn't too far from the answer of $6.90$). You could argue similarly for expression 3 that the answer should be quite close to 1 (it's about 1.26—not quite as close as I would have thought).

Expression 5 is quite a challenge. $2^8=256$, so the eight root of 17.5 you would reason should be quite close to 1, so how would you ever predict that $17.5^{\frac{5}{8}}\approx5.98$?

Or perhaps there's some way of algebraically manipulating these expressions to make estimating easier?

Answer 1

Hint: $\quad5^4=625>623,\quad\dfrac25>\dfrac26=\dfrac13,\quad5^3=125,\quad2^{10}=1024,\quad3^4=81>80.9,$

$2^4=16<17.5,\quad\dfrac58>\dfrac48=\dfrac12,\quad5^2=25~>~21.4~>~4^2=16.\quad$

Answer 2

$$17.5^{\frac58}=17.5^{\frac12}\cdot 17.5^{\frac18}\approx 4\cdot \sqrt 2 \approx 4\cdot 1.4=5.6$$ For 3, you might try $$\sqrt[10]{10.24}=2\cdot \sqrt[10]{0.01} =\frac2{\sqrt[5]{10}}$$ and as $ \sqrt2^5<\sqrt 2^6=8<10$, this is $<\sqrt 2$.

For 6, $$ 21.4^{\frac32}=21.4\cdot\sqrt{21.4}\approx 20\cdot 5=100.$$

Answer 3

Ok I got a clean one for comparing (2) and (6):

$ 125^{2/5} > 17.5^{5/8}$ because

$ 5^{6/5} > 17.5^{5/8}$ because

$ 5 > (17.5)^{25/48}$ because

$17.5 < 5^2 \cdot 5^{-2/25}$ because (keeping in mind that $17.5 = 25\cdot0.7$)

$5^{2/25} < 1/0.7 $ because $(1/0.7)^2 > 2$ and

$5^{4/25} < 5^{1/6} < 2 $ because

$5 < 2^6$.