Simplify $\sqrt{28x} \cdot \sqrt{15x} \cdot \sqrt{21x}$. Express your answer in simplest radical form in terms of $x$.

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}".
Writing everything in terms of prime factorizations, the given expression is \[\sqrt{7 \cdot 2^2 \cdot 5 \cdot 3 \cdot 3\cdot 7 \cdot x^3} = \sqrt{(2^2 \cdot 3^2 \cdot 7^2 \cdot x^2) \cdot (5 \cdot x)} = \boxed{42x\sqrt{5x}}.\]
42x\sqrt{5x}
Level 5
Algebra
Problem: Simplify $\sqrt{28x} \cdot \sqrt{15x} \cdot \sqrt{21x}$. Express your answer in simplest radical form in terms of $x$.

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}".
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To simplify the expression, 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:

1. Combine the square roots: 
   $\sqrt{28x} \cdot \sqrt{15x} \cdot \sqrt{21x} = \sqrt{(28x)(15x)(21x)}$.

2. Calculate the product inside the square root:
   - First, multiply the coefficients: $28 \cdot 15 \cdot 21$.
   - Break down the multiplication:
     - $28 = 4 \cdot 7$,
     - $15 = 3 \cdot 5$,
     - $21 = 3 \cdot 7$.
   - Thus, $28 \cdot 15 \cdot 21 = (4 \cdot 7) \cdot (3 \cdot 5) \cdot (3 \cdot 7)$.
   - Combine like terms: $4 \cdot 3^2 \cdot 5 \cdot 7^2$.

3. Calculate the numerical product:
   - $3^2 = 9$,
   - $7^2 = 49$,
   - Therefore, $4 \cdot 9 \cdot 5 \cdot 49 = 4 \cdot 9 = 36$ and $36 \cdot 5 = 180$.
   - Finally, $180 \cdot 49 = 8820$.

4. Now, include the $x$ terms:
   - The expression becomes $\sqrt{8820x^3}$.

5. Factor $8820$ to simplify:
   - $8820 = 42 \cdot 210 = 42 \cdot (2 \cdot 3 \cdot 5 \cdot 7)$.
   - The prime factorization of $8820$ is $2^2 \cdot 3^2 \cdot 5 \cdot 7^2$.

6. Simplify the square root:
   - $\sqrt{8820} = \sqrt{(2^2)(3^2)(7^2)(5)} = 2 \cdot 3 \cdot 7 \cdot \sqrt{5} = 42\sqrt{5}$.

7. Combine with the $x$ terms:
   - $\sqrt{x^3} = x^{3/2} = x \cdot \sqrt{x}$.

8. Final expression:
   - Combine the results: $42\sqrt{5} \cdot x \cdot \sqrt{x} = 42x\sqrt{5x}$.

Thus, the final answer is $42x\sqrt{5x}$.<|END_OF_LATENT|> The final answer is $42x\sqrt{5x}$.