Enjoy This Flavor (6)

[12rk34]

Which one of the following is greatest one ?

(1) 3^3322 (2) 33^332 © 333^22 (4) 22^333 (5) none of these

[shoebite]

um.. (2)? Obviously (5) is not the answer :P

[supersuj]

..IMO..A

 

1) 3^3322

 

2) 33^332 = (11*3)^332 = 3^332* (3*3)^332 = appx 3^1000

 

3) 333^22 = 111^22*3^22 = (3*3*3*3)^22 = appx 3^100

 

4) 22^333 = (3*3*3)^333 = appx 3^1000

[ankitharish]

A seems to be the answer.

[e.cartman]

If you don't have time to bring them to same index or base, just bet on the highest index :)

[shoebite]

@e.cartman: ah..that makes it easier! But will that always work? Like, I was down to A and B, but then if in the exam I dont have any time to do any more calc, and HAVE to guess, should I always bet on the highest index?

[e.cartman]

NO! It wont always work of course! But whenever I see a flavor of this problem, the answer has always been highest index. But it's still only a bet! That would be my resort if running out of time. (Those familiar with asymptotic behavior will note that a^n will beat n^a in the long run...for high values of n)

[bose]

i got A.

3^3322=3^322*3^3000=3^322*27^1000 which is > 33^332 © 333^22 (4) 22^333

[12rk34]

Explanation :

 

This problem is based on the fact that if a > c and b > d then a^b > c^d.

 

Now 3^3322 and 33^332

i.e. 3^(4x830) x 3^2 and 33^332

i.e. (81)^830 x (3^4)^0.5 and 33^332

i.e. (81)^830.5 > 33^332

So 3^3322 > 33^332 ----- (A)

 

Now 22^333 and 333^22

i.e. 22^ (2 x 166) x 22^(2x0.5) and 333^22

i.e. (22^2) ^166.5 and 333^22

i.e. 484^166.5 > 333^22

So 22^333 > 333^22 ----- (B)

 

Now 33^332 and 22^333

i.e. (3^332) x (11^332) and (2^333) x (11x333)

i.e. {3^(2x166)} x (11^332) and {2^(3 x 111)} x (11x332) x 11

i.e. {9^166 x (11^332)} and (8^111) x (11x332) x 11

So 33^332 > 22^333 ----- ©

So 3^3322 > 33^332 > 22^333 > 333^22

Hence (1). :)

How did you like this flavor, dear friends ?

[kool_sunny]

Explanation :

This problem is based on the fact that if a > c and b > d then a^b > c^d.

 

Now 3^3322 and 33^332

i.e. 3^(4x830) x 3^2 and 33^332

i.e. (81)^830 x (3^4)^0.5 and 33^332

i.e. (81)^830.5 > 33^332

So 3^3322 > 33^332 ----- (A)

 

Now 22^333 and 333^22

i.e. 22^ (2 x 166) x 22^(2x0.5) and 333^22

i.e. (22^2) ^166.5 and 333^22

i.e. 484^166.5 > 333^22

So 22^333 > 333^22 ----- (B)

 

Now 33^332 and 22^333

i.e. (3^332) x (11^332) and (2^333) x (11x333)

i.e. {3^(2x166)} x (11^332) and {2^(3 x 111)} x (11x332) x 11

i.e. {9^166 x (11^332)} and (8^111) x (11x332) x 11

So 33^332 > 22^333 ----- ©

So 3^3322 > 33^332 > 22^333 > 333^22

Hence (1). :)

How did you like this flavor, dear friends ?

 

 

 

Ohhhhh my God....

12rk34 this is simply great.....:)