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: $$\boxed{{\displaystyle 30\sqrt{2}}}$$