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To determine the value of \( a \) so that the terms \( M \) and \( N \) are similar, let's break down the given expressions.

### Expression for \( M \):
\[
M = -25 a x^{a+2}
\]

### Expression for \( N \):
\[
N = \sqrt{2} x^{3a - 8}
\]

For the terms \( M \) and \( N \) to be similar, their variables must have the same powers. This means that the exponents of \( x \) in both expressions must be equal. Therefore, we need to set the exponents of \( x \) equal to each other:

\[
a + 2 = 3a - 8
\]

Now, let's solve for \( a \):

\[
a + 2 = 3a - 8
\]

1. Subtract \( a \) from both sides:
\[
2 = 2a - 8
\]

2. Add 8 to both sides:
\[
10 = 2a
\]

3. Divide both sides by 2:
\[
a = 5
\]

Thus, the value of \( a \) is \( \boxed{5} \).

Would you like further clarification or details?

### Related Questions:
1. What are similar terms in algebra?
2. Why do the exponents of \( x \) need to be equal for terms to be similar?
3. How can we solve equations with multiple variables?
4. What is the significance of coefficients in determining similar terms?
5. Can there be more than one solution for \( a \) in certain cases?

### Tip:
When solving for unknowns in expressions involving powers, always compare the exponents of the variables directly to find relationships between them.

Determine el valor de 'a' si se tiene los términos semejantes M y N:
M = -25 a x^(a+2) 
N = sqrt(2) x^(3a-8)

Learn how to find the value of 'a' by equating exponents in algebraic terms M and N. This step-by-step solution covers exponent manipulation and algebraic techniques, suitable for high school students.

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