Question

What is the product of the following expression in simplest radical form 2 radical 12 times radical 27 times 2radical 25

Answer 1

To simplify the expression \(2\sqrt{12} \times \sqrt{27} \times 2\sqrt{25}\), let's first break down each square root into its prime factors.

1. **Calculating the square roots:**
- \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}\)
- \(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}\)
- \(\sqrt{25} = \sqrt{5^2} = 5\)

2. **Substituting the simplified square roots back into the expression:**
\[
2\sqrt{12} \times \sqrt{27} \times 2\sqrt{25} = 2(2\sqrt{3}) \times (3\sqrt{3}) \times 2(5)
\]
Let's also rearrange the constants and square roots:
\[
= 2 \times 2 \times 2 \times 3 \times 5 \times \sqrt{3} \times \sqrt{3}
\]

3. **Calculating the product of the constants:**
\[
2 \times 2 \times 2 = 8
\]
and,
\[
3 \times 5 = 15
\]
So the constants multiply to:
\[
8 \times 15 = 120
\]

4. **Calculating the product of the square roots:**
\(\sqrt{3} \times \sqrt{3} = \sqrt{3^2} = 3\)

5. **Now, putting everything together:**
\[
120 \times 3 = 360
\]

Thus, the final result in simplest radical form is:
\[
\boxed{360}
\]