The expression you provided is:

\[
(5^{3x - 10})(5^{-8x + 12})
\]

We can simplify this by using the **product rule of exponents**, which states that:

\[
a^m \cdot a^n = a^{m+n}
\]

Here, both terms have the same base \(5\). So, we can add their exponents:

\[
5^{(3x - 10)} \cdot 5^{(-8x + 12)} = 5^{(3x - 10 + (-8x + 12))}
\]

Simplifying the exponent:

\[
3x - 10 + (-8x + 12) = 3x - 8x - 10 + 12 = -5x + 2
\]

Thus, the simplified expression is:

\[
5^{-5x + 2}
\]

### Final answer:
\[
(5^{3x - 10})(5^{-8x + 12}) = 5^{-5x + 2}
\]

Would you like more details or have any further questions?

### 5 Related Questions:
1. How does the product rule of exponents work in other contexts?
2. What are the conditions for using the product rule with different bases?
3. How can we apply similar rules for division of exponents?
4. What happens if we have fractional exponents in the base?
5. How do we handle equations that have both positive and negative exponents?

### Tip:
When dealing with expressions that share the same base, always check if the exponents can be combined using the rules of exponents—this simplifies the problem significantly.

This step-by-step guide walks you through simplifying the exponential expression (5^3x - 10)(5^-8x + 12) using the product rule of exponents. Learn key algebraic concepts, including exponent rules, suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation