Radical Expressions and Radical Functions Section 9.1 Radical Expressions and Radical Functions

Objectives Find square roots Find square roots of expressions containing variables Graph the square root function. Evaluate radical functions Find cube roots Graph the cube root function. Find nth roots

Objective 1: Find Square Roots The number b is a square root of the number a if b2 = a. A radical symbol represents the positive or principal square root of a number. Since 3 is the positive square root of 9, we can write The symbol represents the negative square root of a number. It is the opposite of the principal square root. Since –12 is the negative square root of 144, we can write

Objective 1: Find Square Roots If a is a positive real number, represents the positive or principal square root of a. It is the positive number we square to get a. represents the negative square root of a. It is the opposite of the principal square root of a: . The principal square root of 0 is 0: The number or variable expression under a radical symbol is called the radicand. Together, the radical symbol and radicand are called a radical. An algebraic expression containing a radical is called a radical expression.

EXAMPLE 1 Evaluate each square root: Strategy In each case, we will determine what positive number, when squared, produces the radicand. Why The symbol indicates that the positive square root of the number written under it should be found.

EXAMPLE 1 Evaluate each square root: Solution

Objective 1: Find Square Roots A number such as 81, 225, , and 0.36, that is the square of some rational number, is called a perfect square. Square roots of negative numbers are not real numbers. For example, is not a real number, because no real number squared equals –9. Square roots of negative numbers come from a set called the imaginary numbers. We summarize three important facts about square roots as follows.

Objective 2: Find Square Roots of Expressions Containing Variables Simplifying : For any real number x, The principal square root of x is equal to the absolute value of x. We use this definition to simplify square root radical expressions.

EXAMPLE 2 Strategy In each case, we will determine what positive expression, when squared, produces the radicand. Why The symbol indicates that the positive square root of the number written under it should be found.

EXAMPLE 2 Solution

EXAMPLE 2 Solution

Objective 3: Graph the Square Root Function Since there is one principal square root for every nonnegative real number, the equation determines a function, called a square root function. Square root functions belong to a larger family of functions known as radical functions.

EXAMPLE 3 Graph and find the domain and range of the function. Strategy We will graph the function by creating a table of function values and plotting the corresponding ordered pairs. Why After drawing a smooth curve though the plotted points, we will have the graph.

EXAMPLE 3 Graph and find the domain and range of the function. Solution To graph the function, we select several values for that are perfect squares, such as 0, 1, 4, and 9, and find the corresponding values of f (x). We begin with x = 0, since 0 is the smallest input for which is defined. We enter each value of x and its corresponding value of f (x) in the table. Then we let x = 1, 4, 6, 9, and 16, and list each corresponding function value in a table.

EXAMPLE 3 Graph and find the domain and range of the function. Solution After plotting the ordered pairs, we draw a smooth curve through the points to get the graph shown in figure (a). Since the equation defines a function, its graph passes the vertical line test. (a)

EXAMPLE 3 Graph and find the domain and range of the function. Solution To find the domain of the function graphically, we project the graph onto the x-axis, as shown in figure (b). Because the graph begins at (0, 0) and extends indefinitely to the right, the projection includes 0 and all positive real numbers. To find the range of the function graphically, we project the graph onto the y-axis, as shown in figure (b). Because the graph of the function begins at (0, 0) and extends indefinitely upward, the projection includes all nonnegative real numbers. (b)

Objective 4: Evaluate Radical Functions Radical functions can be used to model certain real-life situations that exhibit growth that eventually levels off.

EXAMPLE 5 Pendulums. The period of a pendulum is the time required for the pendulum to swing back and forth to complete one cycle. The period (in seconds) is a function of the pendulum’s length L (in feet) and is given by . Find the period of the 5-foot-long pendulum of a clock. Round the result to the nearest tenth. Strategy To find the period of the pendulum we will find f(5). Why The notation f(5) represents the period (in seconds) of a pendulum whose length L is 5 feet.

EXAMPLE 5 Pendulums. The period of a pendulum is the time required for the pendulum to swing back and forth to complete one cycle. The period (in seconds) is a function of the pendulum’s length L (in feet) and is given by . Find the period of the 5-foot-long pendulum of a clock. Round the result to the nearest tenth. Solution The period is approximately 2.5 seconds.

Objective 5: Find Cube Roots The number b is a cube root of the real number a if b3 = a. The cube root of a is denoted by . By definition = b if b3 = a. In symbols, we can write: . The number 3 is called the index, 8 is called the radicand, and the entire expression is called a radical. All real numbers have one real cube root. A positive number has a positive cube root, a negative number has a negative cube root, and the cube root of 0 is 0.

Objective 5: Find Cube Roots A number such as 125, , –27, and –8, that is the cube of some rational number, is called a perfect cube. To simplify cube root radical expressions, you need to quickly recognize each of the following natural-number perfect cubes shown in pink. The following property is also used to simplify cube root radical expressions. For any real number x, .

Simplify: EXAMPLE 6 Strategy In each case, we will determine what number or expression, when cubed, produces the radicand. Why The symbol indicates that the cube root of the number written under it should be found.

Simplify: EXAMPLE 6 Solution

Objective 6: Graph the Cube Root Function Since there is one cube root for every real number x, the equation defines a function, called the cube root function. Like square root functions, cube root functions belong to the family of radical functions.

EXAMPLE 7 Consider: a. Graph the function. b. Find its domain and range. c. Graph: Strategy We will graph the function by creating a table of function values and plotting the corresponding ordered pairs. Why After drawing a smooth curve though the plotted points, we will have the graph. The answers to parts (b) and (c) can then be determined from the graph.

EXAMPLE 7 Consider: a. Graph the function. b. Find its domain and range. c. Graph: Solution a. To graph the function, we select several values for x, that are perfect cubes, such as –8, –1, 0, 1, and 8, and find the corresponding values of f (x). The results are entered in the table below.

EXAMPLE 7 Solution Consider: a. Graph the function. b. Find its domain and range. c. Graph: Solution After plotting the ordered pairs, we draw a smooth curve through the points to get the graph shown below:

EXAMPLE 7 Consider: a. Graph the function. b. Find its domain and range. c. Graph: Solution b. From the graph on the top right, we see that the domain and the range of function ƒ are the set of real numbers. Thus, the domain is (–∞, ∞) and the range is (–∞, ∞). c. The graph of is the graph of translated 2 units downward. See the graph at bottom right.

Objective 7: Find nth Roots The nth root of a is denoted by , and The number n is called the index (or order) of the radical. If n is an even natural number, a must be positive or zero, and b must be positive. When n is an odd natural number, the expression , where n > 1 represents an odd root. When n is an even natural number, the expression , where x > 0 represents an even root. The rules for :

Objective 7: Find nth Roots We summarize the definitions concerning If n is a natural number greater than 1 and x is a real number,

EXAMPLE 8 Evaluate: Strategy In each case, we will determine what number, when raised to the fourth, fifth, sixth, or seventh power, produces the radicand. Why The symbols indicate that the fourth, fifth, or sixth root of the number written under it should be found.

EXAMPLE 8 Evaluate: Solution