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6.1 Roots and Radical Expressions To Simplify: -Simplify by finding perfect squares (or cubes, etc.) Examples: 52=25, 5 is a square root of 25 (√25=5) 3 53=125, 5 is a cube root of 125 ( √125=5) 4 54=625, 5 is a fourth root of 625 ( √625=5) So by definition the nth root: If an=b, then a is the nth root of b. *VARIABLES: divide exponent by root to simplify—leave “leftovers” (remainders) inside radical When root is even, and variable exponent is odd in your answer, put that variable/exponent in absolute value bars* Examples: Leave 1 Line for each set: Find each real root 1a) √36 1b) √0.25 2a)√16𝑥 2 2b) √27𝑦 6 3 3 1c) − √64 4 3 1d) √−27 5 2c) √𝑥 20 𝑦 28 2d) √32𝑦10 6.2 Multiplying and Dividing Radical Expressions To Multiply: -Keep numbers outside the radical outside and the numbers inside, inside. -DO NOT MULTIPLY INSIDE NUMBERS—put them both under radical (& factor perfect squares--makes simplifying easier) -Simplify EVERYTHING To Divide: -If same root on numerator/denominator, try to divide inside radical first -Then, simplify all exponents and radicals (not really “dividing”—just simplifying as much as possible) -NO radicals can be in denominator (*rationalize*: multiply top & bottom by a number that will make denominator a perfect square, cube, etc. so the radical can be simplified to a whole number) Examples: Leave 2 lines for each problem: Multiply, then simplify. 1)√8 ∙ √32 3 3 2) √4 ∙ √16 4 4 3) 3√18𝑎9 ∙ √6𝑎𝑏 2 Leave 3 full lines each: Divide, then simplify. 3 √48𝑥 3 𝑦 2 4) 3 √6𝑥 4 𝑦 5) 6) √𝑥 √2 √5𝑥 4 𝑦 √2𝑥 2 𝑦3 6.3 Binomial Radical Expressions To Add/Subtract: -must have like terms/radicals (Similar to like terms w/ variables) *SOMETIMES YOU MUST SIMPLIFY FIRST* 3√3 2√3 -4√3 √3 all like radicals (√3) 3x 2x -4x x all like terms (x) Example: 4√2+ 5√2 -√2= (4+5-1) √2 = 8√2 …just like 4x + 5x – x = (4+5-1)x = 8x When Multiplying: -Remember to FOIL and to multiply things INSIDE radical separate from things OUTSIDE radical When Dividing: -Binomial denominator: multiply by CONJUGATE (change middle sign—like we did with i) **All other exponent rules apply** 6.3a Examples: Leave 1 line for each problem. Add or subtract if possible. 1) 5√6 + √6 3 2) 4√3 + 4√3 3) 6√18 + 3√50 Leave 2 lines for each problem. Multiply. 4) (2+√7)(1+3√7) 5) (√13 + 6)2 6.3b Examples: 6) (5−√11) (5+√11) Leave 3 full lines each: Divide. Rationalize. 1+√3𝑥 7) √6𝑥 8) 5+√3 2−√3 6.4 Rational Exponents 𝑚 𝑛 𝑎 =𝑎 𝑝𝑜𝑤𝑒𝑟 𝑟𝑜𝑜𝑡 𝑛 √𝑎 𝑚 = 𝑎 −𝑚 𝑛 = 𝑚 𝑛 1 𝑝𝑜𝑤𝑒𝑟 𝑎 𝑟𝑜𝑜𝑡 = 𝑛 1 √𝑎𝑚 𝑛 Exponential form: 𝑎 Radical form: √𝑎𝑚 -Multiplying like bases, add exponents; dividing, subtract exponents -Exponent is outside parentheses: MUTLIPLY exponents -More than one thing in parentheses: distribute to ALL -Negative exponents: move to bottom (put under 1 if no numerator) -All of these work backwards too *No fractional exponents in bottom of fractions!! Examples: Simplify. (1 line) 1a. 27 1 3 1 2 1 2 1b. 7 ∙ 21 Write in radical form. (1 line) 2a. 𝑥 1 3 2b. 𝑥 1.5 Write in exponential form. (1 line) 4 3a. √−10 3b. √𝑐 2 Find each product or quotient. (2 lines) 3 4 4a. √6 ∙ √6 6 4b. √𝑥𝑦 4 √4 4c. 4 3 √𝑥 8 𝑦 2 √4 Simplify each number. (2 lines) 5a. 8 2 3 3 5 1 3 5b. (−243) 1 3 5c. 27 27 Write in simplest form. (2 lines) 2 3 6a. (𝑥 )−3 1 3 6b. (−27𝑥 −9 ) 1 2 −2 3 6c. (𝑥 𝑦 )−6 6d. ( 1 𝑥4 −3 𝑦4 12 ) 6.5 Solving Square Root and Other Radical Equations Radical equation: equation that has a variable in a radicand or a variable with a rational exponent Extraneous solution: solution that does not work in original equation when plugged in & checked Steps to Solve: 1) Isolate radical 2) Square, cube, etc. each side or *multiply fractional exponents by reciprocal on BOTH sides* 3) Solve for x [4) **check for extraneous solutions IF variable is on each side of = sign or you have 2+ variables that don't combine.] 6.5a Examples: (Leave 4-5 lines each) 1) √𝑥 + 4 + 6 = 7 2) √4𝑥 + 1 − 5 = 03) 1 3 3) (6𝑥 + 9) − 5 = −2 5 4) 3 √(𝑥 + 1)3 + 1 = 25 6.5b Examples: (Leave 8-10 lines) 5) √𝑥 + 7 − 5 = 𝑥 6) 2 + √𝑥 − 6 = √𝑥 + 10 6.6 Function Operations **AICE writes this as: gf(x) which means the same as above** Examples: Let f(x) = 3x – 2 and g(x) = x2 + 1. Perform each function operation. (Leave 2 lines each problem) (f∙g)(x) (f–g)(x) (f ° g)(x) f(x) + g(x) g(x) – f(x) f(x) – g(x) Let f(x) = 2x and h(x) = x2 + 4 (Leave 4 lines each problem) (f ° h)(1) (h ° f)(-2) (f ° h)(a) Write a formula for fg(x) in each of the following: (Leave 3 lines each problem) f(x) = 3x f(x) = 𝑥−3 2 g(x) = x + 2 g(x) = 5x 6.7 Inverse Relations and Functions Inverse Function (written as f-1) Steps to find inverse: 1. Change f(x) to y 2. Switch x and y 3. Solve for y Find the inverse. Is it a function?: (Leave 4 lines each) 1) y = 2x – 1 2) y = (3x – 4)2 3) y = 2𝑥 5 Graph each relation and its inverse. (Leave room for a graph and 3 lines for work) 4) y = 2x + 8 If f(x) = x + 2, evaluate: (Leave 4 lines) 5) f-1(3) If g(x) = 2(x – 1), evaluate: (Leave 4 lines) 6) g-1(2)