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: \[ \boxed{30\sqrt{2}} \]