Let's simplify the given expression step by step:
Given expression:
[tex]\[ \frac{9 p^{-2}}{3 q^{-3}} \][/tex]
### Step 1: Simplify the coefficients
First, separate the numerical coefficients from the variable expressions:
[tex]\[ \frac{9}{3} \cdot \frac{p^{-2}}{q^{-3}} \][/tex]
Divide the coefficients [tex]\(9\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[ \frac{9}{3} = 3 \][/tex]
So, our expression simplifies to:
[tex]\[ 3 \cdot \frac{p^{-2}}{q^{-3}} \][/tex]
### Step 2: Simplify the variable expressions
Next, we need to simplify the fraction involving the variables [tex]\(p\)[/tex] and [tex]\(q\)[/tex]. Recall the properties of exponents:
[tex]\[ \frac{a^m}{b^n} = a^{m-n} \][/tex]
In this expression, we can move the variables with negative exponents:
[tex]\[ \frac{p^{-2}}{q^{-3}} = p^{-2} \cdot q^3 \][/tex]
Here’s why this works:
- [tex]\( p^{-2} \)[/tex] can be written as [tex]\(\frac{1}{p^2}\)[/tex].
- [tex]\( q^{-3} \)[/tex] can be written as [tex]\(\frac{1}{q^{-3}} = q^3\)[/tex].
Thus:
[tex]\[ \frac{p^{-2}}{q^{-3}} = p^{-2} \cdot q^3 \][/tex]
### Step 3: Combine the simplified coefficient with the variables
Now, we combine [tex]\(3\)[/tex] with [tex]\(q^3 \cdot p^{-2}\)[/tex]:
[tex]\[ 3 \cdot q^3 \cdot p^{-2} \][/tex]
Thus, the simplified expression is:
[tex]\[ \boxed{3 q^3 p^{-2}} \][/tex]
