Question

Show all of your steps for how to simplify 3√5⋅2√10
. Your final answer should be in the 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 will follow these steps:

1. **Multiply the coefficients**: Start by multiplying the numerical coefficients outside the square roots.
\[
3 \cdot 2 = 6
\]

2. **Multiply the radicands**: Now, multiply the radicands inside the square roots.
\[
\sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50}
\]

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

4. **Simplify \(\sqrt{50}\)**: Next, simplify \(\sqrt{50}\). We can break down 50 into its prime factors.
\[
50 = 25 \cdot 2 = 5^2 \cdot 2
\]
Therefore, we have:
\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\]

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

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