Question

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

2√12⋅√27⋅2√25

Answer 1

To simplify the expression \( 2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25} \), we start by breaking down each square root and multiplying the components.

1. **Calculate each square root individually**:
- \( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \)
- \( \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \)
- \( \sqrt{25} = \sqrt{25} = 5 \)

2. **Substitute these values back into the original expression**:
\[
2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25} = 2(2\sqrt{3}) \cdot (3\sqrt{3}) \cdot (2 \cdot 5)
\]

3. **Simplify the coefficients first**:
- Coefficients: \( 2 \cdot 2 \cdot 2 \cdot 5 = 8 \cdot 5 = 40 \)
- Now, the square root terms:
\[
\sqrt{3} \cdot \sqrt{3} = \sqrt{3^2} = 3
\]
Therefore, we simplify the square root terms:
\[
(2\sqrt{3}) \cdot 3\sqrt{3} = 2 \cdot 3 \cdot \sqrt{3} = 6\sqrt{3}
\]

4. **Combine everything**:
Combine the coefficients with the square root term:
\[
40 \cdot 3 = 120
\]
Thus the expression becomes:
\[
120\cdot\sqrt{3}
\]

Finally, the product in simplified radical form is:
\[
\boxed{120\sqrt{3}}
\]