Let's simplify the expression:

\[
M = 2^{2n} + 3 \times 4^n
\]

### Step 1: Simplify the first term
The first term \(2^{2n}\) can be rewritten as:

\[
2^{2n} = (2^2)^n = 4^n
\]

### Step 2: Substitute into the expression
Now substitute \(2^{2n}\) with \(4^n\):

\[
M = 4^n + 3 \times 4^n
\]

### Step 3: Factor the expression
Since both terms contain \(4^n\), we can factor it out:

\[
M = 4^n (1 + 3)
\]

Simplifying further:

\[
M = 4^n \times 4
\]

\[
M = 4^{n+1}
\]

### Final Simplified Expression
The simplified expression is:

\[
M = 4^{n+1}
\]

This expression indicates that the original function \(M\) can be represented as a single exponential term with base 4.

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Would you like to explore more about this result, or do you have any specific questions?

Here are some related questions:

1. What does the expression \(M = 4^{n+1}\) tell us about the growth of \(M\) as \(n\) increases?
2. How does this simplification help in evaluating \(M\) for specific values of \(n\)?
3. Can we represent the original expression using logarithms?
4. How would the derivative of \(M = 4^{n+1}\) with respect to \(n\) look?
5. What is the significance of the factorization in understanding the behavior of the function?

**Tip:** Factoring is a powerful tool for simplifying expressions and solving equations, especially when dealing with exponential terms.

Learn how to simplify the expression M = 2^(2n) + 3 * 4^n using factorization and exponential function techniques. This problem involves analyzing exponential growth and factorization in mathematics. Suitable for students in Grades 10-12.

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