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Non Perfect Squares, Square Roots, and Estimation Textbook: Page 44 – from CD, let’s do 1 2 3 Page 45 – from UA, do 1a Page 49 – do “Not the Only Square Root” Page 51 – do 8 Page 51 - Challenge do 12abc 13ab 1 Rational numbers have a decimal portion that ends (finite) 3.45 or repeats endlessly (infinitely) 87.19363636… others!!!! Irrational numbers have a decimal portion that never ends and never repeats. 7.45932687… others!!!! The square root of a non-perfect square has a decimal portion that never ends and never repeats. Therefore, the square root of a non perfect square is an irrational number. 6 is a non perfect square. Its square root never ends and never repeats so its square root is an irrational number. The exact answer for the square root of non perfect square 6 is 31 ? 47 ? 59 ? __ √6 __ Using your calculator, see that the approximate answer for the √ 6 is 2.4494897… others!!!! 2 non perfect square exact square root approx square root 2 3 5 6 7 8 11 13 27 29 31 others 3 1 4 9 16 25 36 49 64 81 100 … are examples of perfect squares 1 2 3 4 5 6 7 8 9 10 are their square roots 2 3 5 6 7 8 10 11 13 14 15 17 18 19 20… are non perfect squares We can use a calculator to find their approximate square roots and we can estimate to find their roots. View the pattern for the estimation of the square root of 11 and then 13: __ √11 is between which two perfect squares? __ √? < ___ √9 < 3 < __ √ 11 < __ √? ___ √ 11 < ___ √ 16 = ___ √ 11 < = 4 11 is closer to 9 than it is to 16. Therefore, ___ √ 11 is a little more than 3 4 ___ The √ 13 is between which two perfect squares? __ ___ __ √? < √ 13 < √? = ___ √9 < 3 < ___ √ 13 < ___ √ 13 < ___ √ 16 = 4 13 is closer to 16 than it is to 9. Therefore, __ √13 is a little less than 4 Now, you practice the concept, by completing the following: __ __ __ __ √ 26 √ 34 √ 41 √ 53 __ √ 75 __ √ 89 ___ √ 92 ____ √ 103 5 Follow the pattern: ___ ___ The √ 9 is 3, the √900 is 30 ___ ____ The √ 16 is 4, the √1600 is 40 __ ___ The √ 25 is 5, the √2500 is ? __ The √ 36 is ___ the √3600 is ? ___ The √ 49 is ____ the √4900 is ? ___ The √ 64 is ____ the √6400 is ? 9 900 16 1600 25 2500 examples of Perfect Squares. 36 3600 49 4900 64 6400 are 11 1100 13 1300 are examples of Non Perfect Squares. We estimate ____ ___ ___ the √ 1100 and √1300 in the same manner as √11 __ √9 < 3 < ___ √ 11 < ___ √11 < ___ √ 16 4 ___ √900 < 30 < ___ √ 1100 ____ √ 1100 < ____ √ 1600 < 40 ___ The √1100 is a little more than 30 Use the estimation method to find the approximate square root of the following: __ ___ ____ _____ ____ √ 3300 √6500 √ 7700 √ 8200 √9500 6 Important: Example: 8100 has an even set of zeros or its factors are 81 x 100 so __ ____ √ 81 x √ 100 = 9 x 10 = 90 Shortcut: Find the square root of the 81 and add one zero. Challenge: Does the method work for 810? How about 7900? Quick: What is the square root of 100000000? 7