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10.1 Radical Expressions and Graphs

10.1 Radical Expressions and Graphs. Objective 1. Find square roots. Slide 10.1-3. Find square roots.

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10.1 Radical Expressions and Graphs

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  1. 10.1 Radical Expressions and Graphs

  2. Objective 1 Find square roots. Slide 10.1-3

  3. 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. Square Root A number b is a square root of a if b2 = a. Slide 10.1-4

  4. The positiveor principal square rootof a number is written with the symbol The symbol is used for the negative square rootof a number. Find square roots. (cont’d) 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. Radical Sign Radicand Slide 10.1-5

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

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

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

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

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

  10. 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. 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. Rational Square Roots Perfect Squares 25 144 Slide 10.1-11

  11. CLASSROOM EXAMPLE 4 Identifying Types of Square Roots 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

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

  13. In the number nis the index or orderof the radical. Index Radicand Radical sign 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 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

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

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

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

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

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

  19. CLASSROOM EXAMPLE 7 Graphing Functions Defined with Radicals Graph the function by creating a table of values. Give the domain and range. Solution: Domain: [2, ) Range: [0, ) Slide 10.1- 19

  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: Domain: (, ) Range: (, ) Slide 10.1- 20

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

  22. Find nth roots of nth powers. Slide 10.1- 22

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

  24. Find nth roots of nth powers. Slide 10.1- 24

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

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

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

  28. CLASSROOM EXAMPLE 11 Using Roots to Calculate Resonant Frequency 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. Solution: About 325,000 cycles per second. Slide 10.1- 28

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