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Exploring Square Roots: Rationals, Irrationals, and Decimals

Understand square roots, identify rational/irrational roots, find decimal approximations, utilize Pythagorean theorem & distance formula, explore cube/fourth roots.

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Exploring Square Roots: Rationals, Irrationals, and Decimals

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  1. Chapter 8 Section 1

  2. Evaluating Roots Find square roots. Decide whether a given root is rational, irrational, or not a real number. Find decimal approximations for irrational square roots. Use the Pythagorean theorem. Use the distance formula. Find cube, fourth, and other roots. 8.1 2 3 4 5 6

  3. Objective 1 Find square roots. Slide 8.1-3

  4. 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. Find square roots. Square Root A number b is a square root of a if b2 = a. Slide 8.1-4

  5. 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 8.1-5

  6. The statement is incorrect. It says, in part, that a positive number equals a negative number. Find square roots. (cont’d) Slide 8.1-6

  7. Find all square roots of 64. EXAMPLE 1 Finding All Square Roots of a Number Solution: Slide 8.1-7

  8. Find each square root. EXAMPLE 2 Finding Square Roots Solution: Slide 8.1-8

  9. Find the square of each radical expression. EXAMPLE 3 Squaring Radical Expressions Solution: Slide 8.1-9

  10. Objective 2 Decide whether a given root is rational, irrational, or not a real number. Slide 8.1-10

  11. 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 8.1-11

  12. Tell whether each square root is rational,irrational, or not a real number. EXAMPLE 4 Identifying Types of Square Roots 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 8.1-12

  13. Objective 3 Find decimal approximations for irrational square roots. Slide 8.1-13

  14. Find decimal approximations for irrational square roots. Even if a number is irrational, a decimal that approximates the number can be found using a calculator. Slide 8.1-14

  15. EXAMPLE 5 Approximating Irrational Square Roots Find a decimal approximation for each square root. Round answers to the nearest thousandth. Solution: Slide 8.1-15

  16. Objective 4 Use the Pythagorean theorem. Slide 8.1-16

  17. Be careful not to make the common mistake thinking that equals Use the Pythagorean theorem. Many applications of square roots require the use of the Pythagorean formula. If c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the two legs, then Slide 8.1-17

  18. 11 8 ? EXAMPLE 6 Using the Pythagorean Theorem Find the length of the unknown side in each right triangle. Give any decimal approximations to the nearest thousandth. Solution: Slide 8.1-18

  19. 12 ft 5 ft EXAMPLE 7 Using the Pythagorean Theorem to Solve an Application A rectangle has dimensions of 5 ft by 12 ft. Find the length of its diagonal. Solution: Slide 8.1-19

  20. Objective 5 Use the distance formula. Slide 8.1-20

  21. Use the distance formula. Distance Formula The distance between the points and is Slide 8.1-21

  22. Find the distance between and EXAMPLE 8 Using the Distance Formula Solution: Slide 8.1-22

  23. Objective 6 Find cube, fourth, and other roots. Slide 8.1-23

  24. 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 nis the index or orderof the radical. Index Radicand Radical sign Find cube, fourth, and other roots. It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10. Slide 8.1-24

  25. Find each cube root. EXAMPLE 9 Finding Cube Roots Solution: Slide 8.1-25

  26. Find each root. EXAMPLE 10 Finding Other Roots Solution: Slide 8.1-26

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