Why does the n 'fall out' of the square root in the Pythagorean theorem formula?

[Kruger]

I postet the formula below (attachment). I don't know how we can get from the first step to the second. I mean why does that n fall "out" of the square root?

 

[mathman]

Step 1a, let v=u+n², then lower limit of v is n²
Step 1b, let u=v

I presume w is independent of u.

 

[tacman]

Sure, I'd be happy to explain the formula for you. The formula you posted is called the Pythagorean theorem, and it is used to find the length of the hypotenuse (the longest side) of a right triangle. The first step of the formula is a^2 + b^2, which represents the sum of the squares of the two shorter sides of the triangle. The second step, c^2, represents the square of the hypotenuse.

To understand why the n falls out of the square root, we need to look at the definition of a square root. The square root of a number is the number that, when multiplied by itself, gives the original number. In this case, the square root of n is n^1/2, because n^1/2 * n^1/2 = n.

Now, when we square n^1/2, we get n^2/2. This is because when we multiply two numbers with the same base, we add their exponents. So, n^1/2 * n^1/2 = n^(1/2 + 1/2) = n^1 = n.

In the formula, we are taking the square root of both sides to solve for n. When we do this, the square root and the square cancel each other out, leaving us with just n on the right side. This is why the n "falls out" of the square root.

I hope this explanation helps you understand the formula better. Let me know if you have any other questions.