How do you simplify the square root of 275?

To simplify the square root of 275, we need to break it down into its prime factors and see if any of those factors can be taken out of the square root. Let's walk through the process step by step.

Breaking Down the Number

The first step is to find the prime factorization of 275. We can start by dividing 275 by the smallest prime number, which is 5:

• 275 ÷ 5 = 55

Next, we can factor 55:

• 55 ÷ 5 = 11

Now we have:

• 275 = 5 × 5 × 11

Identifying Perfect Squares

In the expression we found, 275 can be rewritten as:

• 275 = 5² × 11

Here, 5² is a perfect square. This is important because when we simplify square roots, we can take the square root of perfect squares out of the radical.

Simplifying the Square Root

Now we can simplify the square root of 275:

• √275 = √(5² × 11)
• √275 = √(5²) × √11
• √275 = 5√11

Final Result

The simplified form of the square root of 275 is:

5√11

This means that while the square root of 275 is approximately 16.583, in its simplest radical form, it is expressed as 5√11. This method of breaking down numbers into their prime factors is a useful technique for simplifying square roots in general.