I'm so confused on distance formula, I know how to calculate it, but getting the final simplified answer is so confusing. Like so on some equations the final answer is just a regular whole number, others is a number with a square root like 3 -/2, or a number square rooted -/3 it's really confusing for me please help and excuse my squart root symbol (-/) I'm confused when I try to simplify if it's just a regular whole number or a number square rooted or what.

I guess what I'm asking is what are the rules of simplifying the final answer in distance formula after you got the main answer

I can understand why the concept of simplifying the distance formula can be confusing. Let me explain how the simplification works.

The distance formula is used to find the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and can be written as:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points, and d represents the distance between them.

To simplify this formula, you need to evaluate the expression inside the square root and determine whether it can be simplified further. Let's break it down step by step:

Step 1: Expand the squared terms
(x₂ - x₁)² = (x₂ - x₁)(x₂ - x₁) = x₂² - 2x₁x₂ + x₁²
(y₂ - y₁)² = (y₂ - y₁)(y₂ - y₁) = y₂² - 2y₁y₂ + y₁²

Step 2: Substitute the expanded terms back into the formula
d = √(x₂² - 2x₁x₂ + x₁² + y₂² - 2y₁y₂ + y₁²)

Step 3: Combine like terms
When you combine the terms inside the square root, you will end up with various coefficients in front of the squared values:
d = √(x₂² + y₂² - 2x₁x₂ - 2y₁y₂ + x₁² + y₁²)

Step 4: Simplify further if possible
At this point, you can determine whether any of the terms inside the square root can be simplified. If the numbers involved are perfect squares (i.e., their square roots result in whole numbers), you can simplify further.

For example, if x₁² + y₁² simplifies to a perfect square, say a², and x₂² + y₂² simplifies to a perfect square, say b², then the simplified formula would be:
d = √(b² - 2x₁x₂ - 2y₁y₂ + a²)

However, if none of the terms are perfect squares that can be further simplified, you would leave the expression as it is.

In summary, when simplifying the distance formula, you first expand the squared terms, then combine like terms, and finally simplify further if any of the terms are perfect squares. If none of the terms can be simplified any further, you leave the expression as it is.