So I have a question which is:
Which is larger?
$$2.2^{3.3} \text{ or } 3.3^{2.2} $$
Now I need to find out with using a calculator but the answer is $3.3^{2.2}$.
The only thing I could think of is rounding.
So you know:
$2^3=8$ and $3^2=9$
I'm interested in seeing if there are other ways just because there might be the possiblity of being asked:
Which is larger?
$$2.5^{3.5} \text{ or } 3.5^{2.5} $$
So if I use the rounding idea, would I just round normally. I would get:
$$ 3^4 \text{ or } 4^3 $$
which shows $3^4$ is larger.
This is the only way I could think of, are there any other ways without using a calculator to determine which is larger?
Best Answer
$$2.2^{3.3}\gtrless3.3^{2.2}$$ $$(2.2^3)^{1.1}\gtrless (3.3^2)^{1.1}$$ $$2.2^3\gtrless 3.3^2$$ $$(2\cdot 1.1)^3\gtrless (3\cdot 1.1)^2$$ $$2^3\cdot 1.1\gtrless 3^2$$ $$8\cdot 1.1\gtrless 9$$ $$8.8<9.$$
So, $$2.2^{3.3}<3.3^{2.2}.$$
Can you find now which number is larger: $2.25^{3.375}$ or $3.375^{2.25}$ ;) ?