https://file.aiquickdraw.com/mathbot/file/1726728633254ken4n8qw.jpg

The expression in the image you provided is:

\[
\frac{10 \times 5^{x+1} + 25 \times 5^x}{3 \times 5^{x+2} + 10 \times 5^{x+1}}
\]

Let me help you simplify it step by step.

### Step 1: Factor the numerator and denominator

#### Numerator:
\[
10 \times 5^{x+1} + 25 \times 5^x = 5^x \times (10 \times 5 + 25) = 5^x \times (50 + 25) = 5^x \times 75
\]

So, the numerator simplifies to:
\[
75 \times 5^x
\]

#### Denominator:
\[
3 \times 5^{x+2} + 10 \times 5^{x+1} = 5^{x+1} \times (3 \times 5 + 10) = 5^{x+1} \times (15 + 10) = 5^{x+1} \times 25
\]

So, the denominator simplifies to:
\[
25 \times 5^{x+1}
\]

### Step 2: Simplify the entire expression

The original expression now becomes:
\[
\frac{75 \times 5^x}{25 \times 5^{x+1}} = \frac{75}{25} \times \frac{5^x}{5^{x+1}}
\]

First, simplify the constants:
\[
\frac{75}{25} = 3
\]

Then simplify the powers of 5:
\[
\frac{5^x}{5^{x+1}} = \frac{1}{5}
\]

So, the entire expression simplifies to:
\[
3 \times \frac{1}{5} = \frac{3}{5}
\]

### Final Answer:
\[
\frac{3}{5}
\]

Would you like any further details or clarifications?

Here are some related questions you might explore:
1. How can we factor expressions involving exponents more efficiently?
2. What are the rules for simplifying powers of the same base?
3. How do we handle division of exponential terms in algebra?
4. Could a different factoring technique be applied here?
5. What happens if there were different bases involved in the exponents?

**Tip**: Always look for common factors in expressions involving exponents before trying other complex algebraic methods!

Simplify the expression: (10 * 5^(x+1) + 25 * 5^x) / (3 * 5^(x+2) + 10 * 5^(x+1))

Learn how to simplify expressions with exponents using the given problem: (10 * 5^(x+1) + 25 * 5^x) / (3 * 5^(x+2) + 10 * 5^(x+1)). This step-by-step solution demonstrates how to factor and simplify exponential expressions efficiently, making it suitable for Grade 9-10 students.

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