How do you simplify √540?

To simplify the square root of 540, we can break it down into its prime factors. Here’s how you can do it step by step:

Step 1: Factor 540

First, we find the prime factors of 540:

• 540 can be divided by 2: 540 ÷ 2 = 270
• 270 can also be divided by 2: 270 ÷ 2 = 135
• Next, divide 135 by 3: 135 ÷ 3 = 45
• Then, divide 45 by 3 again: 45 ÷ 3 = 15
• Finally, divide 15 by 3: 15 ÷ 3 = 5

So, the prime factorization of 540 is:

540 = 2² × 3³ × 5

Step 2: Apply the Square Root

Now, we can simplify the square root:

• From the factorization, we see that 2² and 3³ can be simplified.
• The square root of 2² is 2.
• The square root of 3³ can be simplified to 3 (since 3² is a perfect square) and we have one 3 left over.

Step 3: Combine the Results

Putting it all together, we have:

√540 = √(2² × 3² × 3 × 5) = 2 × 3 × √(3 × 5)

This simplifies to:

6√15

Final Answer

The simplified form of √540 is 6√15.