10.1 Radical Expressions and Graphs

Objective 1 Find square roots. Slide

Find square roots. When squaring a number, multiply the number by itself. To find the square root of a number, find a number that when multiplied by itself, results in the given number. The number a is called a square root of the number a 2. Slide Square Root A number b is a square root of a if b 2 = a.

The symbol, is called a radical sign, always represents the positive square root (except that ). The number inside the radical sign is called the radicand, and the entire expression—radical sign and radicand—is called a radical. The positive or principal square root of a number is written with the symbol Radical Sign Radicand The symbol is used for the negative square root of a number. Slide Find square roots. (cont’d)

The statement is incorrect. It says, in part, that a positive number equals a negative number. Slide Find square roots. (cont’d)

Find all square roots of 64. Solution: Slide Finding All Square Roots of a Number CLASSROOM EXAMPLE 1

Find each square root. Solution: Slide Finding Square Roots CLASSROOM EXAMPLE 2

Find the square of each radical expression. Solution: Slide Squaring Radical Expressions CLASSROOM EXAMPLE 3

Objective 2 Decide whether a given root is rational, irrational, or not a real number. Slide

Deciding whether a given root is rational, irrational, or not a real number. All numbers with square roots that are rational are called perfect squares. Perfect Squares Rational Square Roots A number that is not a perfect square has a square root that is irrational. Many square roots of integers are irrational. Not every number has a real number square root. The square of a real number can never be negative. Therefore, is not a real number. Slide

Tell whether each square root is rational, irrational, or not a real number. Solution: Not all irrational numbers are square roots of integers. For example  (approx ) is a irrational number that is not an square root of an integer. Slide Identifying Types of Square Roots CLASSROOM EXAMPLE 4

Objective 3 Find cube, fourth, and other roots. Slide

Find cube, fourth, and other roots. Finding the square root of a number is the inverse of squaring a number. In a similar way, there are inverses to finding the cube of a number or to finding the fourth or greater power of a number. The nth root of a is written In the number n is the index or order of the radical. Radical sign Index Radicand It can be helpful to complete and keep a list to refer to of third and fourth powers from Slide

Find each cube root. Slide Finding Cube Roots Solution: CLASSROOM EXAMPLE 5

Find each root. Solution: Slide Finding Other Roots CLASSROOM EXAMPLE 6

Objective 4 Graph functions defined by radical expressions. Slide

Square Root Function The domain and range of the square root function are [0,  ). Slide Graph functions defined by radical expressions.

The domain and range of the cube function are ( ,  ). Slide Graph functions defined by radical expressions. Cube Root Function

Graph the function by creating a table of values. Give the domain and range. xf(x)f(x) –2 –1 0 2 Domain: [  2,  ) Range: [0,  ) Slide CLASSROOM EXAMPLE 7 Graphing Functions Defined with Radicals Solution:

Xf(x)f(x) 33 4 Domain: ( ,  ) Range: ( ,  ) Slide CLASSROOM EXAMPLE 7 Graphing Functions Defined with Radicals (cont’d) Graph the function by creating a table of values. Give the domain and range. Solution:

Objective 5 Find nth roots of nth powers. Slide

For any real number a, That is, the principal square root of a 2 is the absolute value of a. Slide Find nth roots of nth powers.

Find each square root. Slide CLASSROOM EXAMPLE 8 Simplifying Square Roots by Using Absolute Value Solution:

If n is an even positive integer, then If n is an odd positive integer, then That is, use absolute value when n is even; absolute value is not necessary when n is odd. Slide Find nth roots of nth powers.

Simplify each root. Slide CLASSROOM EXAMPLE 9 Simplifying Higher Roots by Using Absolute Value Solution:

Objective 6 Use a calculator to find roots. Slide

Use a calculator to approximate each radical to three decimal places. Slide CLASSROOM EXAMPLE 10 Finding Approximations for Roots Solution:

In electronics, the resonant frequency f of a circuit may be found by the formula where f is the cycles per second, L is in henrys, and C is in farads. (Henrys and farads are units of measure in electronics). Find the resonant frequency f if L = 6  and C = 4  About 325,000 cycles per second. Slide CLASSROOM EXAMPLE 11 Using Roots to Calculate Resonant Frequency Solution: