$$ 15x^2y^3 $$z$$ ^2 $$$$ \frac{\placeholder{}}{\placeholder{}} $$$$ 45x^2y^2 $$z$$ 3^{\placeholder{}} $$

It looks like you're trying to simplify an expression involving two terms:

\[
\frac{15x^2y^3z^2}{45x^2y^2z^3}.
\]

Let's go step by step:

### Step 1: Simplify the coefficients
The coefficient of the numerator is $ 15 $, and the coefficient of the denominator is $ 45 $. Simplifying the fraction:
\[
\frac{15}{45} = \frac{1}{3}.
\]

### Step 2: Simplify the variables with exponents

#### For $ x $:
The exponent of $ x $ in both the numerator and the denominator is the same, $ x^2 $. So, they cancel out:
\[
\frac{x^2}{x^2} = 1.
\]

#### For $ y $:
The exponent of $ y $ in the numerator is $ y^3 $, and in the denominator, it is $ y^2 $. Subtract the exponents:
\[
y^{3-2} = y^1 = y.
\]

#### For $ z $:
The exponent of $ z $ in the numerator is $ z^2 $, and in the denominator, it is $ z^3 $. Subtract the exponents:
\[
z^{2-3} = z^{-1} = \frac{1}{z}.
\]

### Final Expression:
After simplifying the expression, we get:
\[
\frac{15x^2y^3z^2}{45x^2y^2z^3} = \frac{1}{3} \cdot y \cdot \frac{1}{z} = \frac{y}{3z}.
\]

So the simplified expression is:
\[
\frac{y}{3z}.
\]

Would you like any further explanation or more details?

### 5 Related Questions:
1. How do you simplify fractions with multiple variables?
2. What are the rules for subtracting exponents in division?
3. How would you approach similar expressions involving negative exponents?
4. How do you deal with more complex expressions with different bases?
5. Can you explain the properties of exponents and how they apply to simplification?

### Tip:
Always simplify coefficients first when dealing with rational expressions—it often makes the rest of the problem easier to manage!

Learn how to simplify the algebraic expression 15x^2y^3z^2 / 45x^2y^2z^3 with step-by-step explanations. This problem involves applying the laws of exponents and simplifying rational expressions. Perfect for students in Grades 8-10.

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