Show all of your steps for how to simplify 3Show all of your steps for how to simplify 3√5 ⋅2 √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: First, we multiply the numerical coefficients (3 and 2). \[ 3 \cdot 2 = 6 \]

2. Multiply the radicals: Next, we will multiply the square roots. When multiplying square roots, we can use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). So we have: \[ \sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} \] Now calculate \( 5 \cdot 10 \): \[ 5 \cdot 10 = 50 \] Therefore, \[ \sqrt{5} \cdot \sqrt{10} = \sqrt{50} \]

3. Combine the results: Now we can write the expression with the coefficient and the radical: \[ 6 \cdot \sqrt{50} \]

4. Simplify the radical: Next, we simplify \( \sqrt{50} \). We can factor \( 50 \) into \( 25 \cdot 2 \): \[ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]

5. Final multiplication step: Substitute back into the equation: \[ 6 \cdot \sqrt{50} = 6 \cdot 5\sqrt{2} = 30\sqrt{2} \]

So the final simplified answer is: \[ \boxed{30\sqrt{2}} \]