How do you simplify the square root of 240?

To simplify the square root of 240, you can break it down into its prime factors. Here’s how you can do it step by step:

Step 1: Factor 240

First, find the prime factorization of 240:

• 240 = 2 × 120
• 120 = 2 × 60
• 60 = 2 × 30
• 30 = 2 × 15
• 15 = 3 × 5

So, the complete factorization is:

240 = 2^4 × 3^1 × 5^1

Step 2: Apply the Square Root

Now, take the square root of each factor:

• The square root of \(2^4\) is \(2^2 = 4\).
• The square root of \(3^1\) is \(\sqrt{3}\).
• The square root of \(5^1\) is \(\sqrt{5}\).

Step 3: Combine the Results

Putting it all together, you get:

√240 = √(2^4 × 3 × 5) = 4√(15)

Final Answer

The simplified form of the square root of 240 is:

4√15