7.1 Radicals and Radical Functions

Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. In order to find a square root of a, you need a number that, when squared, equals a.

Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number. Radicands

Principal and Negative Square Roots If a is a nonnegative number, then is the principal or nonnegative square root of a is the negative square root of a. Principal Square Roots

Find each square root. a. b. Example c. d.

Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form. Perfect Squares

Cube Root The cube root of a real number a is written as Cube Roots

Example Find each cube root. a. b. c. d.

Other roots can be found, as well. The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number. nth Roots

Example Find each root. a. b. c.

If n is an even positive integer, then If n is an odd positive integer, then Finding nth Roots

Example Simplify. Assume that the variables represent any real number. a. b. c. d.

Since every value of x that is substituted into the equation produces a unique value of y, the root relation actually represents a function. The domain of the root function when the index is even, is all nonnegative numbers. The domain of the root function when the index is odd, is the set of all real numbers. Root Functions

x y x y Graph 6 2 Example

x y 1 0 x y Graph Example