show all the steps for how to simply 3√5 ∙ 2√10 your answer should be in simplest radical form

Answer 1

To simplify the expression \( 3\sqrt{5} \cdot 2\sqrt{10} \), we can follow these steps:

Step 1: Multiply the coefficients

First, we'll multiply the coefficients (the numbers outside the square roots):

\[ 3 \cdot 2 = 6 \]

Step 2: Multiply the square roots

Next, we'll multiply the square roots:

\[ \sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} \]

Calculating the product inside the radical:

\[ 5 \cdot 10 = 50 \]

So, we have:

\[ \sqrt{5} \cdot \sqrt{10} = \sqrt{50} \]

Step 3: Simplify the square root

Now we need to simplify \( \sqrt{50} \). We can factor \( 50 \) as follows:

\[ 50 = 25 \cdot 2 \]

Since \( 25 \) is a perfect square, we can simplify:

\[ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]

Step 4: Combine the results

Now we can combine our results from Step 1 and Step 3:

\[ 6 \cdot \sqrt{50} = 6 \cdot 5\sqrt{2} = 30\sqrt{2} \]

Final Answer

Thus, the expression \( 3\sqrt{5} \cdot 2\sqrt{10} \) simplified is

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