We are given the equation:

\[
25^{(n - 1)} = 5^{(n - 1)} - 100
\]

### Step-by-step solution:

1. **Rewrite \( 25 \) as a power of \( 5 \):**
   \[
   25 = 5^2
   \]
   So, the equation becomes:
   \[
   (5^2)^{(n - 1)} = 5^{(n - 1)} - 100
   \]
   By the laws of exponents, \( (a^m)^n = a^{m \cdot n} \), we can simplify the left-hand side:
   \[
   5^{2(n - 1)} = 5^{(n - 1)} - 100
   \]

2. **Let \( x = 5^{(n - 1)} \):**
   Substituting \( x \) into the equation gives:
   \[
   x^2 = x - 100
   \]

3. **Rearrange the equation:**
   Move all terms to one side:
   \[
   x^2 - x + 100 = 0
   \]

4. **Solve the quadratic equation:**
   Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = 100 \):
   \[
   x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(100)}}{2(1)}
   \]
   \[
   x = \frac{1 \pm \sqrt{1 - 400}}{2}
   \]
   \[
   x = \frac{1 \pm \sqrt{-399}}{2}
   \]
   Since \( \sqrt{-399} \) is an imaginary number, the solutions for \( x \) are complex:
   \[
   x = \frac{1 \pm i\sqrt{399}}{2}
   \]

5. **Interpretation of solutions:**
   Since \( x = 5^{(n - 1)} \), \( x \) must be a real number because the base of the exponentiation is a positive real number. Therefore, there are no real solutions to this equation.

### Conclusion:
The given equation has no real solutions for \( n \).

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you solve quadratic equations with imaginary solutions?
2. What are the laws of exponents, and how are they applied?
3. How does the quadratic formula work for real and complex solutions?
4. Can we solve similar equations involving logarithms?
5. What are the rules for simplifying exponents when the base is expressed as a power?

### Tip:
Always check if the equation involves complex solutions, especially when dealing with square roots of negative numbers.

Explore how to solve the exponential equation 25^(n - 1) = 5^(n - 1) - 100 by applying the laws of exponents and solving a quadratic equation with complex solutions. This step-by-step solution is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation