### Instruction: Calculate $\sqrt{10p} \cdot \sqrt{5p^2} \cdot \sqrt{6p^4}$ . Express your answer in simplest radical form in terms of $p$.

Note: When entering a square root with more than one character, you must use parentheses or brackets.  For example, you should enter $\sqrt{14}$ as "sqrt(14)" or "sqrt{14}".### Response: To simplify this expression, I will use the following properties of square roots:
$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ for any nonnegative numbers $a$ and $b$.
$\sqrt{a^m} = a^{m/2}$ for any nonnegative number $a$ and any integer $m$.
I will start by applying the first property to separate the factors under each square root sign:
$\sqrt{10p} \cdot \sqrt{5p^2} \cdot \sqrt{6p^4} = (\sqrt{10} \cdot \sqrt{p}) \cdot (\sqrt{5} \cdot \sqrt{p^2}) \cdot (\sqrt{6} \cdot \sqrt{p^4})$
Next, I will use the second property to simplify the powers of $p$ under the square roots:
$(\sqrt{10} \cdot \sqrt{p}) \cdot (\sqrt{5} \cdot \sqrt{p^2}) \cdot (\sqrt{6} \cdot \sqrt{p^4}) = (\sqrt{10} \cdot p^{1/2}) \cdot (\sqrt{5} \cdot p) \cdot (\sqrt{6} \cdot p^2)$
Now, I will multiply the coefficients and the powers of $p$ together, using the rules of exponents:
$(\sqrt{10} \cdot p^{1/2}) \cdot (\sqrt{5} \cdot p) \cdot (\sqrt{6} \cdot p^2) = \sqrt{10} \cdot \sqrt{5} \cdot \sqrt{6} \cdot p^{1/2 + 1 + 2}$
$\sqrt{10} \cdot \sqrt{5} \cdot \sqrt{6} \cdot p^{1/2 + 1 + 2} = \sqrt{10 \cdot 5 \cdot 6} \cdot p^{7/2}$
Finally, I will simplify the radical by finding the largest perfect square factor of the radicand:
$\sqrt{10 \cdot 5 \cdot 6} \cdot p^{7/2} = \sqrt{2 \cdot 5 \cdot 3 \cdot 2 \cdot 5} \cdot p^{7/2}$
$\sqrt{2 \cdot 5 \cdot 3 \cdot 2 \cdot 5} \cdot p^{7/2} = \sqrt{2^2 \cdot 5^2 \cdot 3} \cdot p^{7/2}$
$\sqrt{2^2 \cdot 5^2 \cdot 3} \cdot p^{7/2} = (2 \cdot 5) \cdot \sqrt{3} \cdot p^{7/2}$