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 can follow these steps:

1. **Combine the constants**:
\[
2 \cdot 2 = 4.
\]

2. **Combine the square roots**:
\[
\sqrt{12} \cdot \sqrt{27} \cdot \sqrt{25} = \sqrt{12 \cdot 27 \cdot 25}.
\]

3. **Calculate the product under the square root**:
First, we find \( 12 \), \( 27 \), and \( 25 \) in terms of their prime factors:
- \( 12 = 2^2 \cdot 3 \)
- \( 27 = 3^3 \)
- \( 25 = 5^2 \)

Thus, we can compute the product:
\[
12 \cdot 27 \cdot 25 = (2^2 \cdot 3) \cdot (3^3) \cdot (5^2).
\]

Combine these:
\[
= 2^2 \cdot 3^{1+3} \cdot 5^2 = 2^2 \cdot 3^4 \cdot 5^2.
\]

4. **Taking the square root**:
Using the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \):
\[
\sqrt{12 \cdot 27 \cdot 25} = \sqrt{2^2} \cdot \sqrt{3^4} \cdot \sqrt{5^2} = 2 \cdot 3^2 \cdot 5 = 2 \cdot 9 \cdot 5.
\]

Now, calculating \( 2 \cdot 9 \cdot 5 \):
\[
2 \cdot 9 = 18,
\]
\[
18 \cdot 5 = 90.
\]

5. **Final product**:
Therefore,
\[
\sqrt{12 \cdot 27 \cdot 25} = 90.
\]
Now, putting it all together:
\[
4 \cdot 90 = 360.
\]

Thus, the final product of the expression \( 2 \sqrt{12} \cdot \sqrt{27} \cdot 2 \sqrt{25} \) is:
\[
\boxed{360}.
\]