Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lesson 2.7 Finding Square Roots and Compare Real Numbers Objective: You will find square roots and compare real numbers. Why? So you can find side lengths of geometric shapes. EXAMPLE 1 Find square roots 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. EXAMPLE 1 for Example 1 Find square roots GUIDED PRACTICE Evaluate the expression. 1. 2. – 9 25 = –3 = The negative square roots of 9 is – 3. 5 The positive square root of 25 is 5. The positive and negative square root of 64 as 8 and – 8. The negative square roots of 81 is – 9. 3. –+ 64 = –+ 8 4. – 81 = –9 EXAMPLE 2 Approximate a square root 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. EXAMPLE 2 Approximate a square root Number 28 29 30 31 32 Square of number 784 841 900 961 1024 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 Write a compound inequality that compares 945 with both 900 and 961. 900 < 945 < 961 Take positive square root of each number. 30 < 945 < 31 Find square root of each perfect square. EXAMPLE 2 Approximate a square root 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. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE Approximate the square root to the nearest integer. 5. 32 You can use a table to determine whether 32 is a perfect square. Number Square of number 5 6 7 8 25 36 49 64 As shown in the table, 32 is not a perfect square. The greatest perfect square less than 32 is 25. The least perfect square greater than 25 is 36. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE 25 < 32 < 36 Write a compound inequality that compares 32 with both 25 and 36. 25 < 32 < 36 Take positive square root of each number. 5 < 32 < 6 Find square root of each perfect square. Because 32 is closer to 36 than to 25, 32 is closer to 6 than to 5. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE Approximate the square root to the nearest integer. 6. 103 You can use a table to determine whether 103 is a perfect square. Number 8 9 10 11 12 Square of number 64 81 100 121 144 As shown in the table, 103 is not a perfect square. The greatest perfect square less than 103 is 100. The least perfect square greater than 100 is 121. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE 100< 103< 121 Write a compound inequality that compares 103 with both 100 and 121. 100 < 103 < 121 Take positive square root of each number. 10 < 103 < 11 Find square root of each perfect square. Because 100 is closer to 103 than to 121, 103 is closer to 10than to 11. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE Approximate the square root to the nearest integer. 7. – 48 You can use a table to determine whether 48 is a perfect square. Number –6 –7 –8 –9 Square of number 36 49 64 81 As shown in the table, 48 is not a perfect square. The greatest perfect square less than 48 is 36. The least perfect square greater than 48 is 49. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE – 36 < – 48 < – 49 Write a compound inequality that compares 103 with both 100 and 121. – 36 < – 48 < – 49 Take positive square root of each number. – 6 < – 48 < –7 Find square root of each perfect square. Because 49is closer than to 36, – 48 is closer to – 7 than to – 6. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE Approximate the square root to the nearest integer. 8. – 350 You can use a table to determine whether 48 is a perfect square. Number – 17 – 18 – 19 – 20 Square of number 187 324 361 400 As shown in the table, 350 is not a perfect square. The greatest perfect square less than – 350 is – 324. The least perfect square greater than – 350 is – 361. EXAMPLE 2 for Example Approximate a square 2root GUIDED PRACTICE – 324 < – 350 < – 361 Write a compound inequality that compares – 350 with both – 324 and – – 361. – 324 < – 350< – 361 Take positive square root of each number. – 18 <– 350 < – 19 Find square root of each perfect square. Because 361 is closer than to 324, – 350 is closer to – 19 than to – 18. EXAMPLE 3 Classify numbers 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 . Number Real Number? Rational Number? Irrational Whole Number? Integer? Number? 24 Yes No Yes No No 100 Yes Yes No Yes Yes 81 Yes Yes No Yes No EXAMPLE 4 Graph and order real numbers Order the numbers from least to greatest: 4 ,– 5 , 13 , 3 –2.5 , 9 . SOLUTION Begin by graphing the numbers on a number line. ANSWER Read the numbers from left to right: –2.5, – 5 , 4 , 9 , 13 . 3 EXAMPLE 4 fororder Examples 3 and Graph and real numbers GUIDED PRACTICE 4 9. Tell whether each of the following numbers. A rational number,an irrational number, an integer, or a whole – 20 . There order the number: – 9 ,5.2, 0, 7 , 4.1, 2 number from least to greatest. SOLUTION Begin by graphing the numbers on a number line. 9 20 = 4.4 – 2 0 7 = 2.6 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 4.1 5.2 EXAMPLE 4 fororder Examples 3 and Graph and real numbers GUIDED PRACTICE 4 Read the numbers from left to right: 9 – 20 ,– , 0 , 7 , 4.1 , 5.2. 2 Number Real Number? Rational Number? Irrational Number? Integer? Whole Number? – 20 Yes No Yes No No 9 2 Yes Yes No No No 0 No No No No No 7 Yes No Yes No No 4.1 Yes No Yes Yes No 5.2 Yes No Yes Yes No – EXAMPLE 5 Rewrite a conditional statement in if-then form 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 integers are irrational numbers. If-then form: If a number is an integer, then it is not an irrational number. The statement is true. EXAMPLE 5 b. Rewrite a conditional statement in if-then form 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. EXAMPLE 5 Example 5statement in if-then form Rewrite afor conditional 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. 10. All square roots of perfect squares are rational number. SOLUTION Given: 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 irrational number. The statement is true. EXAMPLE 5 Example 5statement in if-then form Rewrite afor conditional 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 repeating decimals are irrational number. SOLUTION Given: All repeating decimals are rational numbers. If-then form: If a number repeating decimals , then it is an irrational number. The statement is false. For example, 0.333 is a repeating decimals can be written as a rational number. EXAMPLE 5 Example 5statement in if-then form Rewrite afor conditional 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. No integers are irrational number. SOLUTION Given: 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.