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10.1 Radical Expressions and Graphs

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Objective 1 Find square roots. Slide 10.1-3

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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 10.1-4 Square Root A number b is a square root of a if b 2 = a.

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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 10.1-5 Find square roots. (cont’d)

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The statement is incorrect. It says, in part, that a positive number equals a negative number. Slide 10.1-6 Find square roots. (cont’d)

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Find all square roots of 64. Solution: Slide 10.1-7 Finding All Square Roots of a Number CLASSROOM EXAMPLE 1

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Find each square root. Solution: Slide 10.1-8 Finding Square Roots CLASSROOM EXAMPLE 2

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Find the square of each radical expression. Solution: Slide 10.1-9 Squaring Radical Expressions CLASSROOM EXAMPLE 3

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Objective 2 Decide whether a given root is rational, irrational, or not a real number. Slide 10.1-10

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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 25 144 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 10.1-11

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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. 3.14159) is a irrational number that is not an square root of an integer. Slide 10.1-12 Identifying Types of Square Roots CLASSROOM EXAMPLE 4

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Objective 3 Find cube, fourth, and other roots. Slide 10.1-13

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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 1-10. Slide 10.1-14

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Find each cube root. Slide 10.1-15 Finding Cube Roots Solution: CLASSROOM EXAMPLE 5

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Find each root. Solution: Slide 10.1-16 Finding Other Roots CLASSROOM EXAMPLE 6

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Objective 4 Graph functions defined by radical expressions. Slide 10.1- 16

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Square Root Function The domain and range of the square root function are [0, ). Slide 10.1- 17 Graph functions defined by radical expressions.

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The domain and range of the cube function are ( , ). Slide 10.1- 18 Graph functions defined by radical expressions. Cube Root Function

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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 10.1- 19 CLASSROOM EXAMPLE 7 Graphing Functions Defined with Radicals Solution:

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Xf(x)f(x) 0 1 2 33 4 Domain: ( , ) Range: ( , ) Slide 10.1- 20 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:

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Objective 5 Find nth roots of nth powers. Slide 10.1- 21

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For any real number a, That is, the principal square root of a 2 is the absolute value of a. Slide 10.1- 22 Find nth roots of nth powers.

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Find each square root. Slide 10.1- 23 CLASSROOM EXAMPLE 8 Simplifying Square Roots by Using Absolute Value Solution:

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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 10.1- 24 Find nth roots of nth powers.

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Simplify each root. Slide 10.1- 25 CLASSROOM EXAMPLE 9 Simplifying Higher Roots by Using Absolute Value Solution:

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Objective 6 Use a calculator to find roots. Slide 10.1- 26

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Use a calculator to approximate each radical to three decimal places. Slide 10.1- 27 CLASSROOM EXAMPLE 10 Finding Approximations for Roots Solution:

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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 10 -5 and C = 4 10 -9. About 325,000 cycles per second. Slide 10.1- 28 CLASSROOM EXAMPLE 11 Using Roots to Calculate Resonant Frequency Solution:

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