The given expression is:

$\frac{15p^7}{(3p^3)^2}$

Let's simplify this step by step.

Step 1: Simplify the denominator

The denominator is $(3p^3)^2$. To simplify this, apply the power of a product rule:

$(3p^3)^2 = 3^2 \cdot (p^3)^2 = 9p^6$

So, the expression becomes:

$\frac{15p^7}{9p^6}$

Step 2: Simplify the fraction

Now simplify the constants and the powers of $p$:

$\frac{15}{9} = \frac{5}{3}$

For the powers of $p$, subtract the exponent in the denominator from the exponent in the numerator:

$p^7 / p^6 = p^{7-6} = p^1 = p$

Final Expression:

After simplification, the final expression is:

$\frac{5p}{3}$

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How would you simplify $\frac{20p^8}{(4p^2)^3}$?
2. What is the general rule for simplifying $\frac{x^m}{x^n}$?
3. How can you simplify expressions with more complex powers like $\frac{18p^{12}}{(6p^4)^2}$?
4. How would you handle an expression where the denominator has a negative exponent, e.g., $\frac{12x^5}{(2x^{-2})^3}$?
5. What happens when the numerator and denominator have the same exponent, like $\frac{p^4}{p^4}$?

Tip: When simplifying expressions involving exponents, remember the quotient rule: $\frac{a^m}{a^n} = a^{m-n}$, which allows you to subtract exponents directly when the bases are the same.