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Real Numbers Subsets, Properties, Order of Operations, Radical Simplification 1 Subsets of Real Numbers Natural Number Set Rational Number Set N: {1, 2, 3, 4, ….} Q: { Whole Number Set These are Fractions! W: {0, 1, 2, 3, 4, ….} Decimal #’s that repeat or Integer Number Set p : where q 0 and p, q Z q } terminate. Z: {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, ….} Irrational Number Set Q c :{ Real numbers that are not rational; nonterminating, non-repeating decimals} For example: 2 2 1.4142135623 731 The Real Number System 3 4 Properties of Real Numbers Let a, b, and c be real numbers. Property Closure Commutative Associative Identity Inverse Addition Multiplication a b is a real # a b is a real # abba ab ba (a b) c a (b c) (ab)c a(bc) a 0 a, 0 a a a 1 a, 1 a a a ( a ) 0 a 1 1, a 0 a The following property involves both addition and multiplication Distributive 5 a(b c) ab ac, (b c)a ba ca Identify the real number property that is being demonstrated… 3 (2 5) (3 2) 5 1 2 1 2 606 2 3 3 2 2 (3 4) (2 3) 4 2(4 7) 2 4 2 7 6 1 1 1 3 3 7 Order of Operations Recall that the four basic math operations are addition, subtraction, multiplication, and division. There is an order in which operations need to be carried out. Order of Operations: (1) Grouping Symbols: [], () (2) Exponents (3) Multiplication/Division from left to right (4) Addition/Subtraction from left to right 8 Original Problem Take care of any exponents Start with parenthesis, inner most parenthesis first 5 Continue with parenthesis, follow multiply/divide and add/subtract rules. Multiply/Divide from left to right Add/Subtract from left to right Simplify, Show all of your steps and label the steps 9 Original problem 18 3 3 2 4 3 3 18 3 32 7 3 18 3 9 7 3 18 3 2 3 18 1 3 6 1 5 10 Inner most parenthesis Exponent Inner most parenthesis Parenthesis Multiply/divide left to right Add/subtract left to right Simplify 11 Anatomy of a Root Radical Sign n 12 Index of the Radical: “The root you are taking” Radicand What it means to find an nth root When finding an nth root, you are actually finding the base of an exponential expression. When evaluating exponential expressions the base grows really big really fast SO When finding nth roots the radicand gets really small really fast 81 9 because 9(9) 81 3 13 27 3 because 3(3)(3) 27 Nth Roots Key properties n x n x, when n is odd n x n x , when n is even Absolute value is not needed if variables are assumed to be positive 14 x x2 x3 x4 x5 x6 x7 x8 x9 x10 1 1 1 1 1 1 1 1 1 1 2 4 8 16 32 64 3 9 27 81 243 729 2187 6561 4 16 64 256 1024 4096 5 25 125 625 3125 15625 78125 6 36 216 1296 7776 46656 7 49 343 2401 15 16807 128 256 512 1024 16384 65535 19683 Simplify 16 Radical Simplification In order to simplify a radical you need to pull any perfect nth root factors that are in the radicand using the following rule: a b a b Simplifying works out much better if this is the biggest perfect nth root factor! (some radicands will have more than one perfect nth root factor) Take the nth root of a and leave b under the radical sign. 17 Real Nth Roots Let n be an integer greater than 1 and let a be a real number. If n is odd, then a has one real nth root: n a If n is even and a > 0, then a has two real nth roots: If n is even and a = 0, then a has one nth root: n 0 0 If n is even and a < 0, then a has no real nth roots. 18 n a Odd nth roots and negative radicands A negative radicand with an odd nth root can be simplified. 3 8 2 2 2 2 5 1 1 1 1 1 1 1 CAUTION! You can not simplify a negative radicand with an even nth root! 19 Writing a radical in simplest radical form Example 50 25 2 55 2 5 2 20 Simplify 21 Simplifying radicals with variables x 3 x x 2 x x x x 22 2 x xx x x Simplify 12 x 3 5 2 x 2 x 9x 6 y 3 12 x 3 9 x6 4 3 x2 x 9 x2 x2 x2 2 3x x 3xxxy y 2 x 3x 3x 3 y y y3 y2 y 23 Combining like objects How many oranges do you have? 24 How many apples do you have? 25 Combining like terms is just like combining like objects. You just need to tell how many of a particular variable you have. How many x’s are there? 26 Combining like terms 27 6x 9x 12 y 7 y 6x 9x 12 y 7 y How many apples and oranges do you have? 28 What to do when the variables aren’t the same 6x 9x 3y 6 x 12 y 7 y 6x 9x 3y 6 x 12 y 7 y 29 30 Adding/Subtracting Like Radicals Combining like radicals is just like combining like terms. Add the coefficient and bring the radical along for the ride. 3 57 5 9 2 7 2 10 5 2 2 In order to add/subtract radical you have to have the same radicand. 31 Unlike Radicals… If you don’t have like radicands then you have to simplify where possible and then add/subtract like radicals 2 8 2 4 2 32 4 3 27 4 3 9 3 22 2 4 3 3 3 3 2 3 33 Simplify Even Roots of Negative Numbers Recall in the last unit we talked about finding nth roots of negative radicands and you were told you couldn’t take the nth root of a negative radicand if the nth root was even. NOW we will explore how to simplify even nth roots of negative radicands. The imaginary unit i is a complex number whose square is 1. i 1 And use this idea to simplify even nth roots of negative radicands So we can say that For Example: 81 81 1 9i 8 1 4 2 1 2 2 2 i2 2 2i 2 34 35