Question

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

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