1 5 Square Roots and Real Numbers Objectives

1 -5 Square Roots and Real Numbers Objectives Evaluate expressions containing square roots. Classify numbers within the real number system. Holt Algebra 1

1 -5 Square Roots and Real Numbers Vocabulary square root perfect square real numbers natural numbers whole numbers integers rational numbers Holt Algebra 1 terminating decimal repeating decimal irrational numbers

1 -5 Square Roots and Real Numbers A number that is _______to form a product is called a square root of that product. The operations of squaring and finding a square root are _______ The radical symbol , is used to represent square roots. Positive real numbers have two square roots. 4 4 = 42 = 16 (– 4) = (– 4)2 = 16 Holt Algebra 1 – =4 Positive square root of 16 = – 4 Negative square root of 16

1 -5 Square Roots and Real Numbers The nonnegative square root is represented by. The negative square root is represented by –. A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table. 0 1 4 02 12 22 32 Holt Algebra 1 9 16 25 36 49 64 81 100 42 52 62 72 82 92 102

1 -5 Square Roots and Real Numbers Reading Math The expression does not represent a real number because there is no real number that can be multiplied by itself to form a product of – 36. Holt Algebra 1

1 -5 Square Roots and Real Numbers Example 1: Finding Square Roots of Perfect Squares Find each square root. A. 42 = 16 =4 B. 32 = 9 = – 3 Holt Algebra 1 Think: What number squared equals 16? Positive square root positive 4. Think: What is the opposite of the square root of 9? Negative square root negative 3.

1 -5 Square Roots and Real Numbers Example 1 C: Finding Square Roots of Perfect Squares Find the square root. Think: What number squared equals 25 ? 81 Positive square root Holt Algebra 1 positive 5. 9

1 -5 Square Roots and Real Numbers Check It Out! Example 1 Find the square root. 1 a. Think: What number squared equals 4? 1 b. Think: What is the opposite of the square root of 25? . Holt Algebra 1

1 -5 Square Roots and Real Numbers The square roots of many numbers like , are not whole numbers. A calculator can approximate the value of as 3. 872983346. . . Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers. Holt Algebra 1

1 -5 Square Roots and Real Numbers Example 2: Problem-Solving Application As part of her art project, Shonda will need to make a square covered in glitter. Her tube of glitter covers 13 square inches. What is the greatest side length Shonda’s square can have? 1 Understand the problem The answer will be the side length of the square. List the important information: • The tube of glitter can cover an area of 13 square inches. Holt Algebra 1

1 -5 Square Roots and Real Numbers Example 2 Continued 2 Make a Plan The side length of the square is because = 13. Because 13 is not a perfect square, is not a whole number. Estimate to the nearest tenth. Find the two whole numbers that is between. Because 13 is between the perfect squares 0 and 16. is between and , or between 3 and 4. Holt Algebra 1

1 -5 Square Roots and Real Numbers Example 2 Continued Because 13 is closer to 16 than to 9, is closer to 4 than to 3. 3 4 You can use a guess-and-check method to estimate. Holt Algebra 1

1 -5 Square Roots and Real Numbers Example 2 Continued 3 Solve is greater than 3. 6. Guess 3. 6: 3. 62 = 12. 96 too low Guess 3. 7: too high 3 3. 72 = 13. 69 3. 6 is less than 3. 7 Because 13 is closer to 12. 96 than to 13. 69, is closer to 3. 6 than to 3. 7. Holt Algebra 1 4 3. 6

1 -5 Square Roots and Real Numbers Example 2 Continued 4 Look Back A square with a side length of 3. 6 inches would have an area of 12. 96 square inches. Because 12. 96 is close to 13, 3. 6 inches is a reasonable estimate. Holt Algebra 1

1 -5 Square Roots and Real Numbers All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics. Holt Algebra 1

1 -5 Square Roots and Real Numbers Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are whole numbers and their opposites: – 3, – 2, – 1, 0, 1, 2, 3, … Rational numbers can be expressed in the form a , b where a and b are both integers and b ≠ 0: 1, 7, 9. 2 1 10 Holt Algebra 1

1 -5 Square Roots and Real Numbers Terminating decimals are rational numbers in decimal form that have a finite number of digits: 1. 5, 2. 75, 4. 0 Repeating decimals are rational numbers in decimal form that have a block of one or more digits that repeat continuously: 1. 3, 0. 6, 2. 14 Irrational numbers cannot be expressed in the a form. They include square roots of whole b numbers that are not perfect squares and nonterminating decimals that do not repeat: , , Holt Algebra 1

1 -5 Square Roots and Real Numbers Example 3: Classifying Real Numbers Write all classifications that apply to each Real number. A. – 32 B. 5 Holt Algebra 1

1 -5 Square Roots and Real Numbers Check It Out! Example 3 Write all classifications that apply to each real number. 4 3 a. 7 9 3 b. – 12 3 c. Holt Algebra 1

1 -5 Square Roots and Real Numbers Lesson Quiz Find each square root. 1. 2. 3. 4. 5. The area of a square piece of cloth is 68 in 2. How long is each side of the piece of cloth? Round your answer to the nearest tenth of an inch. Write all classifications that apply to each real number. 6. 1 7. – 3. 89 8. Holt Algebra 1