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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.

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Evaluate the expression.

EXAMPLE 1 Find square roots Evaluate the expression. a. – + 36 = 6 The positive and negative square are 6 and – 6. roots of 36 b. 49 = 7 The positive square root of 49 is 7. The negative square root of 4 is – 2. c. 4 – = 2

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Evaluate the expression.

EXAMPLE 1 GUIDED PRACTICE Find square roots for Example 1 Evaluate the expression. – 1. 9 = 3 The negative square is – 3. roots of 9 2. 25 = 5 The positive square root of 25 is 5. The positive and negative square root of 64 as 8 and – 8. 64 3. – + = 8 The negative square is – 9. roots of 81 – 4. 81 = 9

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Approximate a square root

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 2 = You need to find the side length s of the tabletop such that s This means that s is the positive square root of 945. You can use a table to determine whether 945 is a perfect square.

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Approximate a square root Number 28 29 30 31 32

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. < 961 900 945 Take positive square root of each number. 30 945 < 31 Find square root of each perfect square.

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Approximate a square root

EXAMPLE 2 Approximate a square root Because 945 is closer to 961 than to 900, is closer to 31 than to 30. 945 ANSWER The side length of the tabletop is about 31 inches.

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 Approximate the square root to the nearest integer. 5. 32 You can use a table to determine whether 32 is a perfect square. Square of number Number 5 6 7 8 64 49 36 25 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.

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 25 < < 36 Write a compound inequality that compares 32 with both 25 and 36. < 36 25 32 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, is closer to 6 than to 5. 32

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 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.

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 100< 103< 121 Write a compound inequality that compares 103 with both 100 and 121. < 121 100 103 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, is closer to 10than to 11. 103

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 Approximate the square root to the nearest integer. 7. 48 – You can use a table to determine whether 48 is a perfect square. Square of number Number – 6 – 7 – 8 – 9 81 64 49 36 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.

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 – 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 < –7 48 – Find square root of each perfect square. Because 49is closer than to 36, – is closer to – 7 than to – 6. 48

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 Approximate the square root to the nearest integer. 8. 350 – You can use a table to determine whether 48 is a perfect square. Square of number Number – 17 – 18 – 19 – 20 400 361 324 187 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.

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Approximate a square root for Example 2

GUIDED PRACTICE Approximate a square root for Example 2 – 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, – is closer to – 19 than to – 18. 350

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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 81 100 No Yes Real Number? Whole Number? Integer? Irrational Number? Rational Number? Number 24 100 81

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EXAMPLE 4 Graph and order real numbers , Order the numbers from least to greatest: 4 3 – 5 13 –2.5 9 . SOLUTION Begin by graphing the numbers on a number line. 4 3 ANSWER Read the numbers from left to right: –2.5, – 5 9 13 . ,

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EXAMPLE 4 GUIDED PRACTICE Graph and order real numbers for Examples 3 and 4 Tell whether each of the following numbers. A rational number,an irrational number, an integer, or a whole number: ,5.2, 0, , 4.1, There order the number from least to greatest. 9. 7 9 2 – 20 SOLUTION Begin by graphing the numbers on a number line. 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 9 – 20 4.4 = 7 2.6 5.2 4.1

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Graph and order real numbers for Examples 3 and 4

GUIDED PRACTICE Graph and order real numbers for Examples 3 and 4 9 2 – Read the numbers from left to right: , 7 20 4.1 5.2. Integer? Number Real Number? Rational Number? Irrational Number? Whole Number? – 20 Yes No Yes No No 9 2 – Yes Yes No No No 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

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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.

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EXAMPLE 5 Rewrite a conditional statement in if-then form b. 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, is a real number but not a rational number. 2

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EXAMPLE 5 GUIDED PRACTICE Rewrite a conditional statement in if-then form for 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. All square roots of perfect squares are rational number. 10. 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.

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EXAMPLE 5 GUIDED PRACTICE Rewrite a conditional statement in if-then form for 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. All repeating decimals are irrational number. 11. 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, is a repeating decimals can be written as a rational number.

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EXAMPLE 5 GUIDED PRACTICE Rewrite a conditional statement in if-then form for 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. No integers are irrational number. 12. 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.

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