I can't for the life of me figure out the sqare root of 75.

8.660254

It's an irrational number. There is no exact square root of 75.

Approximately:

8.6602540378443864676372317075294

To 32 digits (Windows Calculator)

i can't for the life of me figure out the sqare root of 75.

How are you expected to work it out?

There are a couple of different algorithms. The simplest one (the one that is used for computers), you take a guess:

1. Given: N (75 in this case)
2. Guess: 8.5 (because 8*8 is 64 and 9*9 is 81)

3. Divide 75 by 8.5 = 8.8235
4. Average the guess with the dividend in #3 (8.5 + 8.8235) / 2 = 8.66176
5. Use the value obtained in #4 as the new divisor (the new guess) in item #3. (75 / 8.66176 = 8.65874).

repeat until the averaged value is the same as the guess within the desired accuracy.

The second algorithm seems more complicated -- but I learned it in junior high school. I started to describe it and realized that I'd need a lot more space and time to describe it in a manner that would be understood. I then googled this link.

http://www.homeschoolmath.net/teaching/square-root-algorithm.php

If you really want to know this algorithm, look at this link.

Here's another example.

http://www.homeschoolmath.net/teaching/square-root-algorithm-example.php

To simplify try:

\sqrt{75}=\sqrt{3\cdot 25}=\sqrt{3}\cdot 5

Now, multiply the square root of 3 b by 5.

1.732(5)\approx 8.66