Transcript lesson 2.7
2.7 Find Square Roots and Compare Real Numbers Warm Up Lesson Presentation Lesson Quiz 2.7 Warm-Up Find the square of the number. 1. 14 ANSWER 196 2. 16 ANSWER 256 Complete the statements using <, >, or =. 3. – 2.5 ? – 19 8 < ANSWER 4. 5 21 ? 6 25 ANSWER < 2.7 Warm-Up 5. A square room has a side length of 25 feet. What is its area? ANSWER 625 ft2 2.7 Example 1 Evaluate the expression. a. +– 36 b. 49 c. – 4 = + –6 The positive and negative square roots of 36 are 6 and – 6. =7 The positive square root of 49 is 7. = –2 The negative square root of 4 is – 2. 2.7 Guided Practice Evaluate the expression. = –3 1. – 9 2. 25 3. –+ 64 = –+ 8 4. – 81 = –9 = 5 2.7 Example 2 FURNITURE The top of a folding table is a square whose area is 945 square inches. Approximate the side length of the tabletop to the nearest inch. SOLUTION You need to find the side length s of the tabletop such that s2 = 945. This means that s is the positive square root of 945. You can use a table to determine whether 945 is a perfect square. 2.7 Example 2 As shown in the table, 945 is not a perfect square. The greatest perfect square less than 945 is 900. The least perfect square greater than 945 is 961. 900 < 945 < 961 900 < 945 < 961 30 < 945 < 31 Write a compound inequality that compares 945 with both 900 and 961. Take positive square root of each number. Find square root of each perfect square. 2.7 Example 2 Because 945 is closer to 961 than to 900, 945 is closer to 31 than to 30. ANSWER The side length of the tabletop is about 31 inches. 2.7 Guided Practice Approximate the square root to the nearest integer. 5. 32 6 6. 103 10 7. – 48 –7 8. – 350 – 19 2.7 Example 3 Tell whether each of the following numbers is a real number, a rational number, an irrational number, an integer, or a whole number: 24 , 100 , – 81 . Irrational Whole Number? Integer? Number? Real Number? Rational Number? 24 Yes No Yes No No 100 Yes Yes No Yes Yes – 81 Yes Yes No Yes No Number 2.7 Guided Practice 9 – 9. Tell whether is a real number, a rational number, 2 an irrational number, an integer, or a whole number. ANSWER real number and rational number 2.7 Example 4 Compare 4 and 13 . 3 SOLUTION Graph the numbers on a number line. ANSWER Because 4 3 is to the left of 13,. 4 3 < 13 . 2.7 Guided Practice 10. Copy and complete using < or >: (a) – 5 –2.5 ? ANSWER > (b) 7 ? ANSWER < 14 5 2.7 Example 5 Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. SOLUTION a. Given: No fractions are irrational numbers. If-then form: If a number is a fraction, then it is not an irrational number. The statement is true. 2.7 b. Example 5 Given: All real numbers are rational numbers. If-then form: If a number is a real number, then it is a rational number. The statement is false. For example, 2 is a real number but not a rational number. 2.7 Guided Practice Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. 11. All square roots of perfect squares are rational numbers. If-then form: If a number is the square root of perfect square, then it is a rational number. The statement is true. 2.7 Guided Practice Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. 12. All repeating decimals are irrational numbers. If-then form: If a number is a repeating decimal, then it is an irrational number. The statement is false. For example, 0.333… is a repeating decimal and can be written as 1 , so it is 3 a rational number. 2.7 Guided Practice Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample. 13. No integers are irrational numbers. If-then form: If a number is an integer, then it is not an irrational number. The statement is true. 2.7 Lesson Quiz Evaluate the expression. 1. +– 289 ANSWER 2. +– 17 – 36 ANSWER –6 Approximate the square root to the nearest integer. 3. – 21 ANSWER 4. –5 620 ANSWER 25 2.7 5. Lesson Quiz A square courtyard has an area of 272 square feet. What is the side length of the courtyard to the nearest foot? ANSWER 16 ft