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Let's analyze the given mathematical expression step by step. The expression is:

\[
\frac{3^5}{3^9} + 2^{n^2} + \sqrt[2]{\sqrt{36}} - \frac{2^4}{3} + (1^{n^n} - 5)^3
\]

We will simplify it by applying the laws of exponents and roots as required.

### Step 1: Simplify \(\frac{3^5}{3^9}\)
Using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\):

\[
\frac{3^5}{3^9} = 3^{5-9} = 3^{-4} = \frac{1}{3^4} = \frac{1}{81}
\]

### Step 2: Simplify \(2^{n^2}\)
This part is already in its simplest form since it involves an unknown variable \(n\).

### Step 3: Simplify \(\sqrt[2]{\sqrt{36}}\)
We first simplify \(\sqrt{36}\), which is \(6\), and then take the square root of \(6\):

\[
\sqrt[2]{\sqrt{36}} = \sqrt[2]{6} = \sqrt{6}
\]

### Step 4: Simplify \(- \frac{2^4}{3}\)
\[
2^4 = 16, \quad \text{so} \quad - \frac{2^4}{3} = - \frac{16}{3}
\]

### Step 5: Simplify \((1^{n^n} - 5)^3\)
Since \(1^{n^n} = 1\) for any \(n\), this simplifies to:
\[
(1 - 5)^3 = (-4)^3 = -64
\]

### Final Expression
Now, putting everything together, we have:

\[
\frac{1}{81} + 2^{n^2} + \sqrt{6} - \frac{16}{3} - 64
\]

This is the simplified form of the expression. 

Do you want further details or have any questions? Here are some related questions you could explore:

1. What are the rules for simplifying expressions involving exponents and roots?
2. How do we handle negative exponents when simplifying?
3. What is the difference between roots of a number, such as square root and cube root?
4. How would the expression change if the variable \(n\) were given a specific value?
5. Can you apply the same methods to simplify more complex functions, like trigonometric or logarithmic functions?

**Tip**: When simplifying fractions involving powers, always try to apply exponent rules to make the process faster.

Teniendo la siguiente expresión, resuélvela aplicando las leyes de la potenciación y radicación según sea necesario, evidenciando el paso a paso de aplicación de las leyes: (3^5 / 3^9) + 2^(n^2) + sqrt(sqrt(36))^2 - (2^4 / 3) + (1^(n^n) - 5)^3

This tutorial explains how to simplify algebraic expressions involving exponents and roots step by step. We explore exponent rules such as a^m / a^n and radical simplifications like sqrt(sqrt(36))^2. Perfect for Grades 10-12 students.

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