# Congruence with numbers with exponents on top of exponents

1. Oct 14, 2010

### morrowcosom

1. The problem statement, all variables and given/known data

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)

2. Relevant equations

3. 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 web site 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)) ?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 14, 2010

### cyby

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

3. Oct 14, 2010

### cyby

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

4. Oct 14, 2010

### 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?

5. Oct 15, 2010

### QuarkCharmer

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

6. Oct 15, 2010

### 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.)

7. Oct 15, 2010

### 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

8. Oct 15, 2010

### cyby

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