The concept of substitution can be illustrated like this. If A=B, and A=C, that means that B=C. Because of that, when faced with equations, one can replace variables in order to find an answer. For example, if y=2x+3 and y=-1/2x+7, then 2x+3=-1/2x+7. Then one can solve the equation as one would a usual two sided variable equation: and 1/2x to both sides and subtract 3 from both sides too. Then you are left with 3 and 1/2x=7. You can divided both sides by three and a half, and get a final answer of x=2. Then you can plug in the 2 in equation: y=2*2+3, or y=7. Then you have the solution: (2,7).

The problem for me with this concept was that all problems aren't nearly as simple, and require some transforming to make them work. The transforming left me behind at first, but I was able to catch-up and now can easily solve substitution problems.
 
 
One situation where the Pythagorean Theorem would came in handy, would be if you didn't know what diagonal size your computer or television screen was, and you wanted to sell it online. You could replace "a" with the length, and "b" with the width, let's say 3' and 4'. Then you would know that your TV screen in fact consisted of a Pythagorean Triple, and was 5' in diagonal length. Also, if the world was being overrun by killer triangles, derived from a 4 meter square, then you would know that the triangle had legs of 4 meters and a hypotenuse of about 5.65 or the square root of 32 (4^2+4^2=c^2). Then you would have more information about your enemy and would be able to defeat them easier.

The Pythagorean Theorem has many possible uses, but its main one right now, is so the Pre-Algebra students can pass their ACS test.
 
 
A square root is the number that can be squared to make the original number. For example, the square root of 4 is 2. The square root of 100 is 10. Square roots are most likely called that because they are the inverse, or the opposite, of squaring. The word "root" comes in to play when it shows the origin, or the beginning, of the original number. The origin would be the number that would be squared, or the answer. Another name for square roots, or more specifically, roots in general, would be dexponents. Exponents is the general term for squaring or cubing, etc., and d- or de- is a prefix that means the opposite. Another name could be square's origin, which would imply the origin of the squared number.

Square roots are very handy and interesting, and extremely useful in certain situations. For example, because of the processes taken to solve for a variable using the Pythagorean Theorem, one
 
 
Negative exponents are not seen commonly in most math articles, but they are a good way to write something in a type of shorthand. When you put a negative exponent on a number, what you are doing is this. First you take away the negative sign, then you solve as if a normal exponent, and finally you put the product underneath a one. (ex. 2^-2=1/4, 6^-2=1/36). The higher the exponent number, the higher the denominator, and thus the smaller the fraction. If an answer is multiplied by the positive version of the base number, it becomes as if the exponential number was 1 closer to zero. [ex. (6^-3)*6=1/36.)

Negative exponents are no common, but can be very useful in the right situations. They are a good thing to comprehend, because when they do show up, they're bound to be important.