Question

Show all of your steps for how to simplify 3 5–√⋅2 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. **Multiply the coefficients** (the numbers outside the square roots):
\[
3 \cdot 2 = 6
\]

2. **Multiply the radicands** (the numbers inside the square roots):
\[
\sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50}
\]

3. **Combine the results** to write the expression in terms of the new coefficient and the new radical:
\[
3 \sqrt{5} \cdot 2 \sqrt{10} = 6 \sqrt{50}
\]

4. **Simplify \(\sqrt{50}\)**. Notice that \(50 = 25 \cdot 2\), and we know that \(\sqrt{25} = 5\):
\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2}
\]

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

So, the final simplified form of the expression \(3 \sqrt{5} \cdot 2 \sqrt{10}\) is:
\[
\boxed{30 \sqrt{2}}
\]