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Roots and Real Number System. January 7, 2013 Mr. Pearson Accelerated Class. Objectives:. In this Review lesson we will: Evaluate algebraic expressions containing roots. Classify numbers within the real number system. Words to know….

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## Roots and Real Number System

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**Roots and Real Number System**January 7, 2013 Mr. Pearson Accelerated Class**Objectives:**In this Review lesson we will: Evaluate algebraic expressions containing roots. Classify numbers within the real number system.**Words to know…**• Square root - a number which, when multiplied by itself, produces the given number. (Ex. 7² = 49, 7 is the square • root of 49) • Perfect square- any number that has an integer square root. (ex. 100 is a perfect square) , • Cube root - a number that is raised to the third power to form a product is a cube root. (ex 23=8, =2)**Squares**0² = 0 1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 Perfect Square Roots Square Roots First semester, I said you should have memorized through what number?**Are squares and square roots inverses?**Start Root it Square it Result 3 3 5 5 9 9 A square root is the inverse operation of a square!**Do you know your perfect squares?**7 and -7 25 8 and -8 121 196 3 and -3 Are perfect squares only positive numbers?**Square Roots**Positive real numbers have two square roots. Find the square roots of 16. Solution Positive square root of 16 =4 4 4 = 42= 16 = –4 (–4)(–4) = (–4)2= 16 Negative square root of 16 The square roots of 16 are 4 and - 4.**The small number to the left of the root is the index. In a**square root, the index is understood to be 2. In other words, is the same as . A number that is raised to the third power to form a product is a cube root of that product. The symbol indicates a cube root. Since 23 = 8, = 2. Similarly, the symbol indicates a fourth root: 2 = 16, so = 2. Writing Math Cube roots**You try**Find each root. Think: What number squared equals 81? Think: What number squared equals 25? C. Think: What number cubed equals –216? We know that no number times itself can be negative, but what about three numbers? (–6)(–6)(–6) = 36(–6) = –216 = –6 (–6)(–6)(–6) = 36(–6) = –216 = –6**Think: What number squared equals**Think: What number cubed equals You try Finding Roots of Fractions. a. b.**Think: What number squared equals**Think: What number cubed equals You try… A. B. Finding Roots of Fractions. (–6)(–6)(–6) = 36(–6) = –216 = –6**Approximating Square Roots**Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers. Remember If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square and is irrational.**Approximating Square Roots**Approximate to the nearest whole number. Solution Is between 7² and 8². Is 54 closer to 49 or to 64? Does that place its value closer to 7 or to 8?**Let’s practice…**Determine what two consecutive integers each root lies between. Then determine which number it is closer to. Between 2 and 3 Between 4 and 5 Between 4 and 5 Between 5 and 6**Words to know…**• Natural numbers - The counting numbers. (example: 1, 2, 3…) • Whole numbers - The natural numbers and zero.(example: 0, 1,2,3…) • Integers -The whole numbers and their opposites.(ex: …-3,-2,-1,0,1,2,3…) • Rational numbers - Numbers that can be expressed as a fraction (a/b).**Words to know…**• Terminating decimal -Rational numbers in decimal form that have finite (ends) number of digits. (ex 2/5= 0.40 ) • Repeating decimal -rational numbers in decimal form that have a block for one or more digits that repeats continuously. (ex. 1.3=1.333333333) • Irrational numbers - numbers that cannot be expressed as a fraction including square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat.**The real numbers are made up of all rational and irrational**numbers. Reading Math Note the symbols for the sets of numbers. R: real numbers Q: rational numbers Z: integers W: whole numbers N: natural numbers**–32 can be written in the form .**14 is not a perfect square, so is irrational. Classifying Real Numbers Write all classifications that apply to each real number. A. –32 32 1 –32 = – –32 can be written as a terminating decimal. –32 = –32.0 rational number, integer, terminating decimal B. irrational**7 can be written in the form .**67 9 = 7.444… = 7.4 4 9 can be written as a repeating decimal. –12 can be written in the form . Check It Out! Write all classifications that apply to each real number. a. 7 rational number, repeating decimal b. –12 –12 can be written as a terminating decimal. rational number, terminating decimal, integer**10 is not a perfect square, so**is irrational. 100 is a perfect square, so is rational. 10 can be written in the form and as a terminating decimal. Write all classifications that apply to each real number. irrational natural, rational, terminating decimal, whole, integer**A challenge…**• Would you know how to solve this…. -11 -11 x = 5 or -5**A challenge…**• Solve the variable. -3 -3 x = 8 or -8

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