https://file.aiquickdraw.com/mathbot/file/17266964759853oa30l98.jpg

The expression shown in the image is:

\[
\frac{27uv^3}{45u^5v}
\]

Let's simplify this step by step.

### Step 1: Simplify the coefficients
The numbers 27 and 45 have a greatest common divisor of 9. So, simplify:

\[
\frac{27}{45} = \frac{3}{5}
\]

### Step 2: Simplify the variables
- For the \( u \)-terms: \( u^1 \) (in the numerator) and \( u^5 \) (in the denominator). Subtract the powers using the rule \( \frac{u^m}{u^n} = u^{m-n} \):

\[
\frac{u^1}{u^5} = u^{1-5} = u^{-4}
\]

- For the \( v \)-terms: \( v^3 \) (in the numerator) and \( v^1 \) (in the denominator). Again, subtract the powers:

\[
\frac{v^3}{v^1} = v^{3-1} = v^2
\]

### Step 3: Combine the simplified components
The simplified expression is:

\[
\frac{3v^2}{5u^4}
\]

Would you like any additional explanations, or do you have further questions?

Here are five questions to expand on this:
1. How do you handle negative exponents when simplifying fractions?
2. Can you simplify other expressions with multiple variables in a similar way?
3. What happens when the variables in the numerator and denominator are exactly the same?
4. How would you factor the coefficients if the terms were more complex?
5. How does this process change with more variables or higher powers?

**Tip:** Always remember to subtract exponents when dividing powers of the same base.

Simplify the expression (27uv^3) / (45u^5v).

Learn how to simplify the algebraic expression (27uv^3) / (45u^5v) step by step using exponent rules and basic algebra. This solution covers simplification of coefficients, subtraction of exponents, and combining terms. Suitable for students in Grades 8-10.

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