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1 Professor Busken – Square Roots and the Pythagorean Theorem Learning Objectives: A Find the Square Root of a Number B Approximate Square Roots C Use the Pythagorean Theorem Definition 1. The square of a number is the number times itself. For instance, the square of 4 is 16 because 42 or 4 · 4 = 16. Another square of −4 is also 16 because (−4)2 = (−4) · (−4) = 16. Definition 2. The reverse process of squaring is finding a square root. For example, a square root of 16 is 4 because 42 = 16. Another square root of 16 is also −4 because (−4)2 = (−4) · (−4) = 16. Theorem 1. Every positive number has two square roots. For instance, the square roots of 16 are 4 and −4. √ , called a radical sign, to indicate the positive Definition 3. We use the symbol square root. For example, √ 25 = 5 because 52 = 25 and 5 is positive. √ 9 = 3 because 32 = 9 and 3 is positive. Theorem 2 (Square Root of a Number). The square root, is the positive number b whose square is a. In symbols, √ a = b if b2 = a For example, Find the square root. √ 1. 100 √ √ , of a positive number a 36 = 6 since 62 = 36 1. 2. √ 64 2. 3. √ 81 3. 2 Professor Busken – Square Roots and the Pythagorean Theorem 4. √ 5. r 1 4 5. 6. r 49 16 6. 7. r 4 25 7. 8. r 169 100 8. 121 4. 1 4 Definition 4. Numbers like , , 9 and 36 are called perfect squares because their 4 25 square root is a whole number or a fraction. √ A square root such as 21 cannot be written as a whole number or a fraction since 21 is not a perfect square. It can be approximated by estimating, by using a table, or by using a calculator. √ 21 √ 0 0 √ 1 √ 1 4 2 √ 9 3 √ 16 4 √ 25 5 √ 36 6 √ 49 7 √ 64 8 Consider the diagram above. Above the number line, notice that as the numbers under the radical signs increase their value, and thus their √ placement on the number line, increase too. We can use this fact to estimate that 21 is between whole numbers 4 and 5. √ 9. Without a calculator or table, approximate 51 to the nearest whole number. 9. 10. Use a calculator to approximate √ 51 to the nearest one-thousandth. 10. 11. Without a calculator or table, approximate √ 3 to the nearest whole number. 11. 12. Use a calculator to approximate √ 3 to the nearest one-thousandth. 12. 3 Professor Busken – Square Roots and the Pythagorean Theorem a2 + b2 = c2 Theorem 3. If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a2 + b2 = c2 . hypotenuse c Leg a (leg)2 + (leg)2 = (hypotenuse)2 Leg b Without using a calculator, find the unknown length of each right triangle. 13. leg = 6, hypotenuse = 10. 13. 14. leg = 5, hypotenuse = 12. 14. 15. leg = 9, hypotenuse = 12. 15. Using a calculator, find the unknown length of each right triangle. Round your answer to the nearest thousandth. 16. leg = 30, hypotenuse = 72. 16.