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2.7 Square Roots & Comparing Real Numbers

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Square Root — a number times itself to make the number you started with Radicand — the number under the radical symbol Perfect Square — the square of an integer

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Irrational Number — a number that is not rational Real Number — the set of all rational and irrational numbers

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Find square roots EXAMPLE 1: 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. c. 4 – =2 – The negative square root of 4 is – 2. Evaluate the expression.

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

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Approximate a square root 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 2 = You need to find the side length s of the tabletop such that s 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.

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Approximate a square root EXAMPLE 2: Number 2829303132 Square of number 7848419009611024 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 < < 961 900 945 30 945 < < 31

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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. EXAMPLE 2:

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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. Approximate the square root to the nearest integer. 1. 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 GUIDED PRACTICE

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Because 32 is closer to 36 than to 25, is closer to 6 than to 5. 32 25 < 32 < 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. GUIDED PRACTICE

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Number 89101112 Square of number 6481100121144 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. Approximate the square root to the nearest integer. 2. 103 You can use a table to determine whether 103 is a perfect square. GUIDED PRACTICE

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Because 100 is closer to 103 than to 121, is closer to 10 than to 11. 103 100< 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. GUIDED PRACTICE

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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. Approximate the square root to the nearest integer. You can use a table to determine whether 48 is a perfect square. 3. 48 – Square of number Number – 6 – 7 – 8 – 9 81 64 49 36 GUIDED PRACTICE

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Because 49 is closer than to 36, – is closer to – 7 than to – 6. 48 – 36 < – 48 < – 49 Write a compound inequality that compares 103 with both 100 and 121. Take positive square root of each number. Find square root of each perfect square. < < 36 – 48 – 49 – – 6 < < –7–7 48 – GUIDED PRACTICE

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As shown in the table, 350 is not a perfect square. The greatest perfect square less than 350 is 3 24. The least perfect square greater than 350 is 361. Approximate the square root to the nearest integer. You can use a table to determine whether 48 is a perfect square. 4. 350 – Square of number Number – 17 – 18 – 19 – 20 400 361 324 187 GUIDED PRACTICE

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Because 361 is closer than to 324, – is closer to – 19 than to – 18. 350 – 324 < – 350 < – 361 Write a compound inequality that compares – 350 with both – 324 and – – 361. Take positive square root of each number. Find square root of each perfect square. < < 324 – 350 – 361 – < < – 18 350 – – 19 GUIDED PRACTICE

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Classify numbers 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 81 100 No YesNoYes NoYes Real Number ? Yes NoYes Whole Number ? Integer ? Irrational Number ? Rational Number ? Number 24 100 81

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Graph and order real numbers EXAMPLE 4:, 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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SOLUTION Begin by graphing the numbers on a number line. GUIDED PRACTICE 5 43 2 1 0 –1 –2 –3–4–5–6–7 –8 –9 9 2 – 20 4.4 = 7 = 2.6 0 5.2 4.1 Read the numbers from left to right: Order the numbers from least to greatest:

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GUIDED PRACTICE Classify the following numbers as Real, Rational, Irrational, Integer and/or Whole: Real?Rational?Irrational?Integer?Whole? -2.5 4/3

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