SOLUTION: Simplify and assume that the variables can represent any real number. the question is the SQUARE ROOT of this: (x^2+16x+64). The answer given in the back of my book is the ABSO

Algebra ->  Radicals -> SOLUTION: Simplify and assume that the variables can represent any real number. the question is the SQUARE ROOT of this: (x^2+16x+64). The answer given in the back of my book is the ABSO      Log On


   

Question 342902: Simplify and assume that the variables can represent any real number.
the question is the SQUARE ROOT of this: (x^2+16x+64). The answer given in the back of my book is the ABSOLUTE VALUE of (x+8). I guess I'm missing a step. Please help.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28x%5E2+%2B16x+%2B64%29
Whenever you simplify square roots, you look for perfect square factors of the radicand (the expression inside the radical). The key word here is factor. We need to factor the radicand in order to determine if there are any factors which are perfect squares.

So we need to factor the radicand: x%5E2+%2B16x+%2B64 I don't know how good your factoring is. But this expression factors fairly easily. It one of the expressions where you find the factors of the number at the end, 64, which add up to the coefficient in the middle, 16. (Or you can use factoring patterns on this if you know them.) Even though 64 has a large number of factors, only one pair of them add up to 16. And they are 8 and 8. So x%5E2+%2B+16x+%2B64+=+%28x%2B8%29%28x%2B8%29+=+%28x%2B8%29%5E2. As you can see, the entire radicand turns out to be a perfect square!. Now we can proceed with simplifying the square root:
sqrt%28x%5E2+%2B16x+%2B64%29
sqrt%28%28x%2B8%29%5E2%29
The temptation here is to simply say that just like sqrt%2828%5E2%29+=+28, sqrt%28%28x%2B8%29%5E2%29+=+x%2B8. But this is not correct. Square roots are positive (or zero) and we cannot simplfy a square root into an expression that might be negative. And since we do not know what value x might be we do not know what value x+8 might be. We cannot guarantee that x+8 will never be negative. This is the reason for the absolute value! This is what guarantees that the simplified square root cannot ever be negative.
sqrt%28%28x%2B8%29%5E2%29+=+abs%28x%2B8%29

It is probably good to be in the habit of using the absolute value when you are simplifying square roots. When there are no variables involved, the absolute value will go away. For example:
sqrt%2828%5E2%29+=+abs%2828%29+=+28
sqrt%28%28-28%29%5E2%29+=+abs%28-28%29+=+28
sqrt%28%28x%5E2%29%5E2%29+=+abs%28x%5E2%29 And since x%5E2 can never be negative, abs%28x%5E2%29+=+x%5E2.