Show all of your steps for how to simplify 3sqrt(5) * 2sqrt(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 will follow these steps:

1. Multiply the coefficients: The coefficients in this expression are \( 3 \) and \( 2 \). \[ 3 \cdot 2 = 6 \]

2. Multiply the radicals: Next, we will multiply the square roots. We have \( \sqrt{5} \) and \( \sqrt{10} \). \[ \sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} \]

3. Calculate the product inside the square root: Now, we calculate \( 5 \cdot 10 \): \[ 5 \cdot 10 = 50 \] Therefore, we have: \[ \sqrt{5} \cdot \sqrt{10} = \sqrt{50} \]

4. Now we combine the coefficients with the radical: Putting it all together, we have: \[ 3\sqrt{5} \cdot 2\sqrt{10} = 6\sqrt{50} \]

5. Simplifying \( \sqrt{50} \): Next, we simplify \( \sqrt{50} \). We can factor \( 50 \) as follows: \[ 50 = 25 \cdot 2 \] Therefore, we have: \[ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]

6. Substituting back into the expression: Now, we substitute back to our previous result: \[ 6\sqrt{50} = 6 \cdot 5\sqrt{2} = 30\sqrt{2} \]

Thus, the final answer in simplest radical form is: \[ \boxed{30\sqrt{2}} \]