### Instruction: 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}".### Response: To simplify the product of square roots, I can use the property that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ for any nonnegative numbers $a$ and $b$.
This means I can rewrite the given expression as $\sqrt{28x} \cdot \sqrt{15x} \cdot \sqrt{21x} = \sqrt{(28x)(15x)(21x)}$.
Now I can try to factor the expression inside the square root into perfect squares and other factors.
I can start by finding the prime factorization of each number.
28 = 2 $\cdot$ 2 $\cdot$ 7
15 = 3 $\cdot$ 5
21 = 3 $\cdot$ 7
So, $(28x)(15x)(21x) = (2 \cdot 2 \cdot 7 \cdot x)(3 \cdot 5 \cdot x)(3 \cdot 7 \cdot x)$.
Next, I can rearrange the factors and group them into pairs of equal factors.
This will help me identify the perfect squares.
$(2 \cdot 2 \cdot 7 \cdot x)(3 \cdot 5 \cdot x)(3 \cdot 7 \cdot x) = (2 \cdot 2)(3 \cdot 3)(7 \cdot 7)(x \cdot x)(5 \cdot x)$.
Now I can see that the perfect squares are $(2 \cdot 2) = 4$, $(3 \cdot 3) = 9$, $(7 \cdot 7) = 49$, and $(x \cdot x) = x^2$.
The remaining factor is 5x, which is not a perfect square.
So, I can write the expression inside the square root as a product of perfect squares and another factor: $(2 \cdot 2)(3 \cdot 3)(7 \cdot 7)(x \cdot x)(5 \cdot x) = 4 \cdot 9 \cdot 49 \cdot x^2 \cdot 5x$.
Now I can use the property that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ again, but in reverse, to separate the square root of each factor.
$\sqrt{4 \cdot 9 \cdot 49 \cdot x^2 \cdot 5x} = \sqrt{4} \cdot \sqrt{9} \cdot \sqrt{49} \cdot \sqrt{x^2} \cdot \sqrt{5x}$.
For each perfect square, I can take its square root and write it as a whole number or a variable.
$\sqrt{4} \cdot \sqrt{9} \cdot \sqrt{49} \cdot \sqrt{x^2} \cdot \sqrt{5x} = 2 \cdot 3 \cdot 7 \cdot x \cdot \sqrt{5x}$.
Finally, I can multiply the whole numbers and the variable together to get the coefficient of the square root.
$2 \cdot 3 \cdot 7 \cdot x \cdot \sqrt{5x} = 42x \cdot \sqrt{5x}$.