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Chapter 8 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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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 formula. Use the distance formula. Find cube, fourth, and other roots. 1 1 4 4 3 3 2 2 6 6 5 5 8.18.1

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Find square roots. Slide 8.1 - 3

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 8.1 - 4

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The positive or principal square root of a number is written with the symbol. Find square roots. (cont’d) Slide 8.1 - 5 Radical Sign Radicand 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 symbol – is used for the negative square root of a number.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find square roots. (cont’d) Slide 8.1 - 6 The statement is incorrect. It says, in part, that a positive number equals a negative number.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Find all square roots of 64. Solution: Finding All Square Roots of a Number Slide 8.1 - 7

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find each square root. EXAMPLE 2 Solution: Finding Square Roots Slide 8.1 - 8

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Squaring Radical Expressions Slide 8.1 - 9 Find the square of each radical expression. Solution:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Decide whether a given root is rational, irrational, or not a real number. Slide 8.1 - 10

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Deciding whether a given root is rational, irrational, or not a real number. Slide 8.1 - 11 All numbers with square roots that are rational are called perfect squares. Perfect SquaresRational 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.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Identifying Types of Square Roots Slide 8.1 - 12 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.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide 8.1 - 13 Find decimal approximations for irrational square roots.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find decimal approximations for irrational square roots. Slide 8.1 - 14 A calculator can be used to find a decimal approximation even if a number is irrational.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Approximating Irrational Square Roots Slide 8.1 - 15 Find a decimal approximation for each square root. Round answers to the nearest thousandth. Solution:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Slide 8.1 - 16 Use the Pythagorean formula.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 Use the Pythagorean formula. Be careful not to make the common mistake thinking that equals.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Using the Pythagorean Formula Slide 8.1 - 18 Find the length of the unknown side in each right triangle. Give any decimal approximations to the nearest thousandth. 11 8 ? Solution:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using the Pythagorean Formula to Solve an Application Slide 8.1 - 19 A rectangle has dimensions of 5 ft by 12 ft. Find the length of its diagonal. 5 ft 12 ft Solution:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Use the distance formula. Slide 8.1 - 20

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.1 - 21 Use the distance formula. The distance between the points and is

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using the Distance Formula Slide 8.1 - 22 Find the distance between and. Solution:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 Objective 6 Find cube, fourth, and other roots. Slide 8.1 - 23

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 Find cube, fourth, and other roots. Slide 8.1 - 24 Radical sign Index Radicand In, the number n is the index or order of the radical. It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find each cube root. Solution: EXAMPLE 9 Finding Cube Roots Slide 8.1 - 25

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 Finding Other Roots Slide 8.1 - 26 Find each root. Solution:

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