Question

Show all of your steps for how to simplify 3(sqrt(5) ⋅ 2(sqrt(10). Your final answer should be in simplest radical form (not a decimal). Responses that are an answer only with no work shown will not receive credit.(6 points)

Answer 1

To simplify the expression \(3(\sqrt{5} \cdot 2\sqrt{10})\), we'll follow these steps:

1. **Identify the expression**:
\[
3(\sqrt{5} \cdot 2\sqrt{10})
\]

2. **Use the associative property and rearrange**:
We can rearrange the expression inside the parentheses:
\[
= 3 \cdot \sqrt{5} \cdot 2 \cdot \sqrt{10}
\]

3. **Multiply the constants and the square roots separately**:
First, multiply the constants:
\[
3 \cdot 2 = 6
\]

Now, multiply the square roots:
\[
\sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50}
\]

4. **Combine the results**:
Now we can combine the results:
\[
3(\sqrt{5} \cdot 2\sqrt{10}) = 6\sqrt{50}
\]

5. **Simplify \(\sqrt{50}\)**:
\(\sqrt{50}\) can be simplified as follows:
\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\]

6. **Substitute back into the expression**:
Now substitute \( \sqrt{50} \) back into the expression:
\[
6\sqrt{50} = 6 \cdot 5\sqrt{2} = 30\sqrt{2}
\]

Thus, the final answer in simplest radical form is:
\[
\boxed{30\sqrt{2}}
\]