https://file.aiquickdraw.com/mathbot/file/1726992193758rveolgyh.jpg

The expression in the image is:

\[
\frac{27m^2n^3}{-18m^4n^5}
\]

To simplify this, we can break it down into parts:

### Step 1: Simplify the coefficients
The coefficient part is:

\[
\frac{27}{-18} = -\frac{27}{18} = -\frac{3}{2}
\]

### Step 2: Simplify the powers of \( m \)
For \( m \), we apply the rule of exponents: \( \frac{m^a}{m^b} = m^{a-b} \):

\[
\frac{m^2}{m^4} = m^{2-4} = m^{-2}
\]

### Step 3: Simplify the powers of \( n \)
For \( n \), we apply the same rule:

\[
\frac{n^3}{n^5} = n^{3-5} = n^{-2}
\]

### Step 4: Combine everything
Putting all parts together:

\[
-\frac{3}{2} m^{-2} n^{-2} = -\frac{3}{2} \cdot \frac{1}{m^2} \cdot \frac{1}{n^2} = -\frac{3}{2m^2n^2}
\]

### Final Answer:
\[
-\frac{3}{2m^2n^2}
\]

Would you like further details on any step or have questions?

Here are five related questions to explore:
1. How would the result change if the powers of \( m \) and \( n \) were higher?
2. Can this expression be rewritten with positive exponents only?
3. How can this type of simplification be applied in solving equations?
4. What happens if either \( m \) or \( n \) equals zero?
5. How can we generalize the simplification process for any exponents?

**Tip**: When simplifying algebraic fractions, remember to handle coefficients and variables with exponents separately for clarity.

Simplify the expression 27m^2n^3 / -18m^4n^5.

Learn how to simplify the algebraic expression 27m^2n^3 / -18m^4n^5 using exponent rules and fraction simplification. This problem covers key algebraic concepts, including simplifying powers of variables and handling coefficients. Suitable for Grades 8-10.

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