Question

Show all of your steps for how to simplify 3sqrt5⋅2sqrt10

Answer 1

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

1. **Multiply the coefficients** (the numbers outside the square roots):
\[
3 \cdot 2 = 6
\]

2. **Multiply the square roots**:
\[
\sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50}
\]

3. **Combine the results from steps 1 and 2**:
\[
3\sqrt{5} \cdot 2\sqrt{10} = 6 \cdot \sqrt{50}
\]

4. **Simplify \(\sqrt{50}\)**:
- First, factor \(50\):
\[
50 = 25 \cdot 2
\]
- Then, take the square root:
\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\]

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

Thus, the simplified expression is:
\[
\boxed{30\sqrt{2}}
\]