# Congruence with numbers with exponents on top of exponents

• morrowcosom
In summary: The problem is that the last number in the equation (3^2*5^3, which means take 3 to the 2*5 power then take this answer to the third power)) is so huge that it will not fit into a calculator. Is there a way to shrink it down or to figure out how it compares to the mod by looking at it?
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)

## 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)) ?

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

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

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?

morrowcosom said:
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

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.)

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

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

morrowcosom said:
(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.

## 1. What is the meaning of congruence in the context of numbers with exponents on top of exponents?

Congruence is a mathematical term that refers to two numbers or expressions being equal in value. In the context of numbers with exponents on top of exponents, congruence means that the two expressions have the same value despite having different exponents.

## 2. How do you determine if two numbers with exponents on top of exponents are congruent?

To determine congruence between two numbers with exponents on top of exponents, you need to simplify the expressions and compare the resulting values. If the values are equal, then the numbers are congruent.

## 3. Can numbers with different exponents on top of exponents be congruent?

Yes, numbers with different exponents on top of exponents can be congruent. As long as the resulting values are equal, the numbers are considered congruent.

## 4. What are some real-world applications of congruence with numbers with exponents on top of exponents?

Congruence with numbers with exponents on top of exponents is commonly used in fields such as physics, chemistry, and engineering. It is used to simplify complex calculations and equations, making them easier to solve and understand.

## 5. Are there any mathematical rules or properties that apply to congruence with numbers with exponents on top of exponents?

Yes, there are several rules and properties that apply to congruence with numbers with exponents on top of exponents. These include the power rule, the product rule, and the quotient rule. These rules make it easier to determine congruence between expressions with exponents.

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