### Roots of Real Numbers

Finding the square root of a number and squaring a number are inverse operations.

To find the square root of a number n, you must find a number whose square is n.  For example, a square root of 49 is 7 since 72 = 49.  Likewise, (–7)2 = 49, so –7 is also a square root of 49.

#### Definition of Square Root

For any real numbers a and b, if a2 = b, then a is a square root of b.
NOTE:  Every positive real number has two real number square roots.  The number 0 has just one square root, 0 itself.  Negative numbers do not have real number square roots.

#### Understanding the symbols

This symbol represents the principal square root of a.
The principal square root of a non-negative number is its nonnegative square root.
For example:

In other words,

To name the negative square root of a we use this symbol.
For example:

To indicate both square roots, use the plus/minus sign which indicates positive or negative.
For example:

Negative numbers do not have real number square roots.
For example:

#### Terminology

The symbol is called a radical sign.  An expression written with a radical sign is called a radical expression.  The expression written under the radical sign is called the radicand.

#### Cube Roots

Definition of Cube Root

The number c is the cube root of a if its third power is a, that is, c3 = a.  Every real number has exactly one cube root in the real number system.

For example: 2 is the cube root of 8 because 23 = 8, or .

–3 is the cube root of –27 because (–3)3  =  –27, or

#### nth Roots

For any real numbers a and b, and any positive integer n, if an = b, then a is the nth root of b.
##### example:

Whenever the number n in is an odd number, we say we are taking an odd root.  When the index is an even number, we say that we are taking an even root.  The number n is called the index.  When the index is 2 we do not write it.

When we take any odd root of a number, we find that there is just one answer.  If the number is positive, the root is positive.  If the number is negative, the root is negative.

Every positive real number has two nth roots when n is even. One of these roots is positive and one is negative.  Negative real numbers do not have nth roots when n is even.

Absolute value signs are never needed when finding odd roots.  When finding even nth roots, absolute value signs are sometimes necessary, as with square roots.

##### odd index – one answer only, a negative constant, no absolute value bars needed since index is odd.

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