One-Two-Three Go!

Oh yeah! I was trolled. Nice question.

if b!=+1,-1 then is the answer 36?

I disagree with your proposed solution. As I read the problem, it is "a raised to the b power, which is then raised to the c power is equal to 6561." After factoring 6561, you have 6561= 3 raised to the eighth power. The question asks what is the greatest possible value of a times b times c. In your answer, you have not satisfied the initial premise of "a raised to the b raised to the c = 6561. You say that 6561 raised to the first is 6561. I agree. However, now, you must raise 6561 to the (very large number). This is obviously not going to be 6561. For example, just try squaring 6561, and you no longer have 6561. I believe that your error is in assuming that b is your base of the exponential expression. It is not, a is the base.

I may be wrong on my answer, however, I disagree with your answer.

After rereading the question, and seeing other answers, I realized that I was making some assumptions. First, it says that a,b, and c are integers. Okay. Are they unique integers? At first, I assumed that and came up with the following.

I hope we may agree that 6561 = 3 raised to the 8th power. So, we know that a = 3. We also know that b times c equal 8. Factor 8 into 1 and 8 or 2 and 4. It doesn't matter. 3 times 1 times 8 or 3 times 2 times 4 = 24. a raised to the b raised to the c = 6561.

Now, assume that a, b, and c are any integers, not necessarily unique integers. The factors of a number may not be larger than the number, when using integers(possibly redundant). A number raised to the first power is the number itself. Due to the power of the exponentiation, the exponents in the equation will be smaller than the number if expanded, for example, 3 squared = 9, both 3 the base, and 2 the exponent are smaller than 9. Thus when you multiply a b c, it will be largest when a is largest, and b and c are smallest. I offer that a = 6561, b = 1 and c = 1. Multiply and you get 6561.

My explanation may be confusing or wordy. I would appreciate help in the details. However, given the assumptions, I believe that I have the correct answer.

Humbly submitted by someone who has not taken a math class in over 30 years.

ItIts really a beautiful question ......easy yet elegant. Subtle yet dominant.....hahahahahah.....cheers sandeep sir.....keep posting problems for us.....

...dang it. This is a good question! :)

Nice question though

Trapped :(

if b =! (+/-)1 then answer is 162?

In response to abhishek sharma;

Sorry but I could"nt understand your solution .please elaborate :-)

How about a=81, b=2, c=1 ??

I have a doubt why can't it be this a=3,b=2,c=3and we get 3 to the power of 8 = 6561