Congruence with numbers with exponents on top of exponents

[morrowcosom]

Homework Statement

Calculate 2 + 3^2 + 5^3 + 3^2*5^3 in Z15
(That last group of numbers means 3 to the 2*5 power then take this answer to the third power, It just did not paste like that)

Homework Equations

The Attempt at a Solution

2+9+5+3<--the 3 is a random guess, because I am totally lost what do to do with this last number.
I am doing this problem for independent study off of a website and it says the answer is 9, which means the last answer must be 8. How do you deal with congruence when there are such large numbers with multiple exponents like those created by the last sequence of numbers in the problem (3^2*5^3, which means take 3 to the 2*5 power then take this answer to the third power)) ?

 

[cyby]

First off, just to clarify, you do mean $2 + 3^2 + 5^3 + (3^{2*5}) ^ {3} in Z_{15}$, right?

 

[cyby]

If this is so, then you should know at least what $(3^{2*5})^{3}$ evaluates to...

 

[morrowcosom]

My problem is that I cannot figure out what the above number evaluates to congruence wise. It is so huge when you figure it up that it will not fit into a calculator. Is their some procedure to shrink this number or these exponents down to size, or figure out how the number compares to the mod by looking at?

 

[QuarkCharmer]

My problem is that I cannot figure out what the above number evaluates to congruence wise. It is so huge when you figure it up that it will not fit into a calculator. Is their some procedure to shrink this number or these exponents down to size, or figure out how the number compares to the mod by looking at?

The EE button on your calculator might be useful here.
http://mathforum.org/library/drmath/view/54346.html

 

[HallsofIvy]

Just the caculator that comes with "Windows" gives $3^{2*5}= 3^{10}= 59049$, not all that big! Find what that is congruent to modulo 15 and raise that to the 3 rd power.

(For that matter, $\left(3^{10}\right)^3$ is NOT too big to be done exactly on any decent calculator.)

 

[Mentallic]

Maybe he means $$3^{2.5^3}$$ ? This would explain why it's too big for his calculator, because(3¹⁰)³$\approx$10^14 should be doable for any calculator made in this millenium

It is so huge when you figure it up that it will not fit into a calculator.

 

[cyby]

(That last group of numbers means 3 to the 2*5 power then take this answer to the third power, It just did not paste like that)

That's what he said, so I'm assuming that we were on the right track.