The expression to simplify is:

$\frac{3p^2 \times 9p^{-1/2}}{3p^{1/4}}$

Let's simplify it step by step:

Step 1: Simplify constants

We have $3$ in the numerator and denominator, so we can cancel them out. Now the expression becomes: $\frac{p^2 \times 9p^{-1/2}}{p^{1/4}}$

Step 2: Combine powers of $p$

Using the property of exponents $p^a \times p^b = p^{a+b}$, the numerator becomes: $9p^{2 + (-1/2)} = 9p^{3/2}$

Step 3: Divide powers of $p$

Now apply the rule $\frac{p^a}{p^b} = p^{a-b}$ to divide the terms: $\frac{9p^{3/2}}{p^{1/4}} = 9p^{3/2 - 1/4} = 9p^{6/4 - 1/4} = 9p^{5/4}$

Final answer:

The fully simplified expression is: $9p^{5/4}$

Do you want more details or have any questions? Here are some related questions:

1. What happens if the powers in the numerator and denominator are negative?
2. How do you simplify terms with fractional exponents?
3. What is the general rule for multiplying exponents?
4. How do you handle the division of terms with fractional powers?
5. How can we simplify expressions with both constants and variables?

Tip: Always simplify constants first to make exponent manipulation easier!