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The problem provided is:

\[
\text{Simplifica } M = \frac{2^{m+3} + 2^{m+2} - 2^{m+1}}{2^{m+2} + 2^{m}}
\]

Let's simplify this expression step by step:

1. **Factor out common terms in the numerator:**

   \[
   2^{m+3} + 2^{m+2} - 2^{m+1} = 2^{m+1} \left(2^2 + 2^1 - 1\right) = 2^{m+1} \cdot (4 + 2 - 1) = 2^{m+1} \cdot 5
   \]

   Thus, the numerator simplifies to \(2^{m+1} \cdot 5\).

2. **Factor out common terms in the denominator:**

   \[
   2^{m+2} + 2^m = 2^m \left(2^2 + 1\right) = 2^m \cdot 5
   \]

   So, the denominator simplifies to \(2^m \cdot 5\).

3. **Simplify the fraction:**

   \[
   M = \frac{2^{m+1} \cdot 5}{2^m \cdot 5}
   \]

   Canceling out \(2^m\) and \(5\) gives:

   \[
   M = 2
   \]

The simplified form of \(M\) is \(2\).

Would you like further details on any of the steps, or have any questions? Here are some related questions to explore:

1. What are the rules for simplifying expressions with exponents?
2. How can factoring help in simplifying algebraic expressions?
3. What is the significance of cancelling common factors in fractions?
4. How would the solution change if the powers of 2 in the original problem were different?
5. What happens if the base of the exponents was a number other than 2?

**Tip:** When simplifying expressions with exponents, look for common factors that can be factored out to simplify calculations.

Simplifica M = (2^(m+3) + 2^(m+2) - 2^(m+1)) / (2^(m+2) + 2^m)

Learn how to simplify the exponential fraction M = (2^(m+3) + 2^(m+2) - 2^(m+1)) / (2^(m+2) + 2^m) through factoring and exponent rules. This problem covers exponentiation and algebraic fraction simplifications, suitable for students in Grades 9-12.

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