Calculate $\sqrt{30p} \cdot \sqrt{5p} \cdot \sqrt{6p}$ . Express your answer in simplest radical form in terms of $p$.
Writing everything in terms of prime factorizations, the given expression is $\sqrt{2 \cdot 3\cdot 5 \cdot 5 \cdot 2 \cdot 3 \cdot p^3} = \sqrt{(2^2 \cdot 3^2 \cdot 5^2 \cdot p^2) \cdot (p)} = \boxed{30p \sqrt{p}}$.
30p\sqrt{p}
Level 4
Algebra
Problem: Calculate $\sqrt{30p} \cdot \sqrt{5p} \cdot \sqrt{6p}$ . Express your answer in simplest radical form in terms of $p$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the expression $\sqrt{30p} \cdot \sqrt{5p} \cdot \sqrt{6p}$, we first apply the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This allows us to combine the square roots into a single square root:

1. Combine the terms under one square root:
   $\sqrt{30p} \cdot \sqrt{5p} \cdot \sqrt{6p} = \sqrt{(30p) \cdot (5p) \cdot (6p)}$.

2. Calculate the product inside the square root:
   - First, multiply the numerical coefficients: $30 \cdot 5 \cdot 6$.
   - Calculate $30 \cdot 5 = 150$.
   - Then, calculate $150 \cdot 6 = 900$.
   - Next, multiply the $p$ terms: $p \cdot p \cdot p = p^3$.

3. Combine the results:
   The expression simplifies to $\sqrt{900p^3}$.

4. Simplify the square root:
   - Recognize that $900$ is a perfect square: $\sqrt{900} = 30$.
   - For $p^3$, we can express it as $p^2 \cdot p$, where $\sqrt{p^2} = p$.

5. Therefore, we have:
   $\sqrt{900p^3} = \sqrt{900} \cdot \sqrt{p^3} = 30 \cdot p \cdot \sqrt{p}$.

6. Finally, combine the terms:
   The final expression is $30p\sqrt{p}$.<|END_OF_LATENT|> The final answer is $30p\sqrt{p}$.