?Which is larger √2 or 3√3

elationship √2+√3≈π2+3≈π$2+3\approx \pi$ is acoincidence and has no significance even though neat polygons can be drawn around circles to demonstrate this relationship.

Given any arbitrary constant between zero and ten, I suspect you can get quite close to that constant with a small number of small integers and a few basic arithmetic operators. The number of permutations of just the ten digits with say half a dozen operators with up to seven symbols in total would be more than

167=268,435,456167=268,435,456$167=268,435,456$

Commutativity and associativity of operators would significantly reduce the total number of values that could be generated, but just a million of them would, on average, mean that one of these expressions could approximate your arbitraryconstant to five decimal places! (Note that the decimal expansion of the number might be just such an expression, but let's ignore that triviality.)

In the light of this it is no surprise that ππ$\pi$ can be approximated to two decimal places with a simple looking expression involving two digits and three operators.

Having said that, I do not promise to give you a simple expression for your favourite Golden ratio (φ=1.61803…φ=1.61803…$\phi =1.61803\dots$), Euler's number (e=2.71828…e=2.71828…$e=2.71828\dots$), Conway's constant (λ=1.30357…λ=1.30357…$\lambda =1.30357\dots$), or whatever.

Brought to you by the Campaign to Demystify ππ$\pi$: there is nothing mystical about ππ$\pi$.

easy way 
3√3 > √2

Since,
3 > 2
Taking square roots both the sides.
√3 > √2
Multiplying both the sides by 3,
3√3 > 3√2
And now, we know,
3√2 > √2

So this implies,
3√3 > √2