Meaning and Properties of Square Roots

1. Definition of Square Root
1. Square Root
   When a number is squared to become , is called the square root of .
   means that is the square root of .
2. Number of Square Roots
1. A positive number has two square roots: a positive and a negative, and their absolute values are the same.
2. Zero has only one square root, which is .
3. A negative number has no real square root, since squaring any real number always results in a non-negative number.


2. Notation of Square Roots
1. The symbol is used to denote the square root, and it is called the radical sign. It is read as “square root” or simply “root”.
   square root of , root
2. Among the square roots of a positive number , the positive one is called the positive square root, and the negative one is called the negative square root:
   Positive square root : , Negative square root :
3. The positive and negative square roots together can be represented as:
4. When the number inside the square root is the square of a rational number, the square root can be written without the radical:
   Example: Square roots of :
   
   
5. Comparison between "Square Root of " and "Positive Square Root of " (For )

   	Square Root of	Positive Square Root of
   Meaning	A number squared to become	The positive square root of
   Notation	,
   Number of Values



3. Properties of Square Roots
1. For :
1. Squaring the square root of returns :
   ,
2. If the number inside the radical is the square of a number, the square root can be written without the radical:
   ,
2. Property of
   Since is the positive square root of , it is always non-negative, regardless of the sign of :
   
   
3. Perfect Squares
1. Perfect Square : A number that is the square of a natural number, such as
2. If the number inside the radical is a perfect square, the square root can be written as a natural number: