What is the product of the following expression in simplest radical form?

Question

2√12 ⋅ √27 ⋅ 2√25

(1 point)
Responses

48100−−−−√
4 square root of 8100

32
32

360
360

1209–√

Answer 1

To simplify the expression \(2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25}\), we can start by simplifying each radical.

1. Simplifying each radical:

   • \(\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}\)
   • \(\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}\)
   • \(\sqrt{25} = 5\)

2. Substituting the simplified radicals back into the expression: \[ 2\sqrt{12} = 2 \cdot 2\sqrt{3} = 4\sqrt{3} \] \[ 2\sqrt{25} = 2 \cdot 5 = 10 \]

Now, we can rewrite the original expression: \[ 4\sqrt{3} \cdot 3\sqrt{3} \cdot 10 \]

3. Combining the coefficients: \[ 4 \cdot 3 \cdot 10 = 120 \]

4. Combining the radicals: \[ \sqrt{3} \cdot \sqrt{3} = \sqrt{3^2} = 3 \]

5. Now, putting it all together: \[ 120 \cdot 3 = 360 \]

Thus, the product of the given expression in simplest radical form is: \[ \boxed{360} \]