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how does it get from the first step to the next

To explain how the expression moves from one step to the next, let's break down the reasoning.

### Expression Progression:
1. **First step**:  
   \( 5 \times (5^{n-1}) \)

2. **Second step**:  
   \( 5^n \times \left( \frac{4}{5} \right) \)

3. **Third step**:  
   \( 4 \times 5^{n-1} \)

### Transition between first and second steps:
The first step \( 5 \times (5^{n-1}) \) involves multiplying the base 5 by an exponential term with the same base \( 5^{n-1} \). When multiplying terms with the same base, the exponents are added:
\[ 5 \times 5^{n-1} = 5^1 \times 5^{n-1} = 5^{1+(n-1)} = 5^n \]

This gives the first part of the second step, \( 5^n \).

Then, the next factor (outside the parentheses) changes to \( \frac{4}{5} \), likely due to some additional factoring or manipulation done in the problem context. It simplifies the expression for the next step.

### Transition between second and third steps:
In the second step, you have \( 5^n \times \frac{4}{5} \). This can be simplified by splitting the base \( 5^n \) as follows:
\[
5^n \times \frac{4}{5} = \frac{4 \times 5^n}{5} = 4 \times 5^{n-1}
\]
This is how you get the third step, \( 4 \times 5^{n-1} \).

### Summary of transitions:
1. \( 5 \times (5^{n-1}) = 5^n \)
2. \( 5^n \times \frac{4}{5} = 4 \times 5^{n-1} \)

Would you like further details or a clarification on a specific part?

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### Related Questions:
1. How can the property of exponents \( a^m \times a^n = a^{m+n} \) be applied in other cases?
2. What are some common factoring techniques used in simplifying expressions like these?
3. How would the expression change if the base were not the same in both terms?
4. What happens if you multiply or divide by other constants like \( \frac{3}{5} \) instead of \( \frac{4}{5} \)?
5. Can this process be extended to negative exponents or fractional powers?

### Tip:
Always look for common factors in exponential terms when simplifying expressions—it can help streamline complex calculations.

How does it get from the first step to the next?

Understand the step-by-step simplification of exponential expressions, transitioning from 5 × 5^{n-1} to 4 × 5^{n-1}. Learn key algebraic concepts, including the exponent addition rule and simplification techniques. Suitable for students in Grades 9-12.

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