Show all of your steps for how to simplify 3 5–√⋅2 10−−√

Question

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