Let's break down the multiplication of the given expression step by step.
The expression we need to simplify is:
[tex]$4 \sqrt{3} \cdot 10 \sqrt{12} \cdot \sqrt{6} \cdot \sqrt{2}$[/tex]
First, let's rearrange the constants and the square roots:
[tex]$= (4 \cdot 10) \cdot (\sqrt{3} \cdot \sqrt{12} \cdot \sqrt{6} \cdot \sqrt{2})$[/tex]
Calculate the product of the constants:
[tex]$= 40 \cdot (\sqrt{3} \cdot \sqrt{12} \cdot \sqrt{6} \cdot \sqrt{2})$[/tex]
Next, we multiply the square roots together. Recall that the product of square roots is the square root of the product of the numbers:
[tex]$ \sqrt{3} \cdot \sqrt{12} \cdot \sqrt{6} \cdot \sqrt{2} = \sqrt{3 \cdot 12 \cdot 6 \cdot 2}$[/tex]
Now, we calculate the product inside the square root:
[tex]$ 3 \cdot 12 = 36 $[/tex]
[tex]$ 36 \cdot 6 = 216 $[/tex]
[tex]$ 216 \cdot 2 = 432 $[/tex]
Thus,
[tex]$ \sqrt{3 \cdot 12 \cdot 6 \cdot 2} = \sqrt{432} $[/tex]
Simplify [tex]$\sqrt{432}$[/tex]:
Factor 432 into its prime factors:
[tex]$ 432 = 2^4 \cdot 3^3 $[/tex]
Take the square root:
[tex]$ \sqrt{432} = \sqrt{2^4 \cdot 3^3} = 2^2 \cdot 3^{3/2} = 4 \cdot 3 \sqrt{3} = 12 \sqrt{3} $[/tex]
Therefore,
[tex]$4 \sqrt{3} \cdot 10 \sqrt{12} \cdot \sqrt{6} \cdot \sqrt{2} = 40 \cdot 12 \sqrt{3} = 480 \sqrt{3}$[/tex]
So, the simplified form in simplest radical form is:
[tex]$ \boxed{480 \sqrt{3}} $[/tex]
The numerical result from performing this multiplication is approximately:
[tex]$ \approx 831.384387633061 $[/tex]
Thus, the final answers are:
The simplest radical form is [tex]\( 12 \sqrt{3} \)[/tex] and the numerical value of the expression is approximately [tex]\( 831.384387633061 \)[/tex].
