MATH 108 REVIEW TOPIC 6 Radicals

Transcription

1 Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises

2 Math 08 T6-Radicals Page The arithmetic and algebra of radicals play a significant role in mathematics. Radicals are used etensively in trigonometry. Since your instructors will insist on eact answers in simplest form (not a decimal approimation), it is important that you master basic manipulations involving radicals. I. Computations with Radicals Let s start with a brief assessment. Eercise. Epress in simplest form. 6 f) 0 6 g) h) ( 5) 7 i) ( )( +) 0 j) 0 Answers Comments:. Some radicals have an eact root. n a = b if b n = a. In other words, 6 = because =6 6 = because =6 6 = because =6 Question: If and ( ) both equal 6, why is 6 not written as ±? Answer: 6 = but not because we want to avoid ambiguity in simplifying radicals. Only the positive or principle root is written. However, in solving equations, all possible roots that satisfy a given equation must be written. Illustration: =6 = ± 6 = ±.

3 Math 08 T6-Radicals Page. Simplifying refers to factoring the quantity inside a radical and then removing any factor that has an eact root. We ll illustrate simplifying with two slightly different methods: Perfect Square Factors i) ii) 0 = 0 = 0 = 0 75 = 5 =5 ab = a b iii) 0 = 8 5= 8 5= 5 perfect cube Repeated Factors i) 0 = 0 = 0 = 0 ii) 75 = 5 = 5 =5 for a>0, n an = a iii) 0 = 5= 5. Basic algebra concepts still apply. Combining Like Terms 5+5 5=8 5; similar to +5 =8 Eponent Rules ( 5) =( 5) = (5) = 0; similar to (a ) =a = = 0 ( ) ; similar to = 8 y y. 6 Finding Products ( + ) = + ( ) + ( ) ; similar to ( +) = + + =7+ Recall: (a + = a +ab + b See Review Topic if you need help squaring a binomial by inspection.

4 Math 08 T6-Radicals Page II. Radicals Containing Variables Removing variables from inside a radical is essentially the same as removing constants. y = (y) y = y y 6 = ( ) = Remember n ( ) n =( ) Alternate Method: In part IV of this Review Topic we discuss n m = m/n. Thus 6 = 6/ =. +6 += ( +) = + WARNING!!! + + Only repeated factors can be removed from a radical. The reason = is because and are factors. This error is so common it is worth repeating. a b = ab but a + b a+ b. Only (a + = a + b. y 5 y y 8 y = 7 = = y 7 Notice the fraction was simplified before roots were taken. Eercise. Epress in simplest form. 6 f) ( y) g) h) ( ) ( ) y 0 y 5 y i) y j) 5 k) True or False: is = for all values of? Answers

5 Math 08 T6-Radicals Page 5 III. Rationalizing A fraction containing a radical in its denominator is NOT considered to be in simplest form. Removing radicals from the denominator is called rationalizing. Illustration:,, 5, Here is how it is done. Rationalizing = and = = = = = 5 = 5 Have you noticed: can all be simplified by rationalizing. = or = 5 = 5 i) each fraction is being multiplied by a form of... ii) resulting in a denominator of the form n ( ) n, thereby causing the radical to be removed. Rationalizing binomial denominators requires a different approach. Because (a (a + =a b, multiplying by the appropriate binomial factor (still in the form of ) will remove radicals. = + = + = ( +) + ( ) = ( +) = ( )( ) + = ( +) =

6 Math 08 T6-Radicals Page 6 Eercise. Rationalize; epress in simplest form. 6 5 y y y f) Rationalize the numerator for. Answers IV. Rational Eponents What is the meaning of / or / or /? Definition: a /n = n a; a m/n = n a m =( n m Any epression with a rational eponent can be rewritten in radical form. In certain situations this is quite helpful. Illustration: Evaluate /, /,and6 /. Answers. / = = / =( ) = =8 6 5/ =( 6) 5 = 5 = Although a calculator can do these problems, we suggest you only use one to check your results. Doing arithmetic (fractions, eponents, logs, etc.) by hand reinforces basic skills that you need for algebra.

7 Math 08 T6-Radicals Page 7 Some problems that start in radical form are made easier by changing to rational eponents. Eamples: ( ) 8 =( / ) 8 = =8 = / / = 7/ 6 ( ) 6 / y = = y y Eercise. Find the value of each (please no calculators). ( / ( / ( ) / ( ) / 8 8 ( Answers 7 6 Change to rational eponents, then simplify. f) ( ) 6 (g) 5 5 (h) 6 Answers

8 Math 08 T6-Radicals Page 8 V. Logarithms If you need an introduction or review of logorithms, see Review Topic, Section IV. Evaluate: log 8 log 8 Answers: log 8 = log 8 =( ) 8 ( ) =. Since 8=8 / =,( )=. log 8 = log 8 =( ) ( ) = 8. Since =,( )=. 8 Eercise 5. Evaluate the following. log log log 8 log 0 00 log 00 0 f) log 00 0 Answers Beginning of Topic 08 Review Topics 08 Skills Assessment

9 Math 08 T6 Eercise Answers Page Epress in simplest form f) ( 5) 0 g) ( )( +) 0 h) 0 Answers: 0 0 Not a real number. 5 5 f) (5) = 5 g) ( ) = h) 0 = 5 Return to Review Topic

10 Math 08 T6 Eercise Answers Page 0 Epress in simplest form. g) 6 5 y ( ) ( ) h) ( 5 y (f) ( y) y 0 y (i) 0 +5 j) 5 (k) True or False: is = for all values of? Answers: (8 ) = 8 or 6 =( 6 ) / = 8 ( ) = or =( ) / = = 8 = ( ( y )(6y) =y 6y (7y )( )=y (f) y + y g) 6 = 6 = (h) y 8 = / y / = y 8 / i) ( 5) = 5 (j) Can t be simplified; No repeated factors. k) False. This statement is true as long as is non-negative; 5 =5or 00 = 00, and so on. But what if = 5? = ( 5) = 5 = 5 =, not. This leads to a more precise definition: =. This is true for all. In calculus you will be epected to use this definition. Return to Review Topic

11 Math 08 T6 Eercise Answers Page Rationalize. Consider all variables positive. 6 5 y y y f) Rationalize the numerator for. Answers: f) 6 = 6 = = 6 = y y y = y y 6 5+ = ( 5+) 5 5+ ( 5) =( 5+) y + y = ( y)( + y) = + y y + y y + = + ( )( +) = + Return to Review Topic

12 Math 08 T6 Eercise Answers Page Find the value of each (please no calculators). / / ( 8 7 ( 8 6 ) / ) / Change to rational eponents, then simplify. f) ( ) 6 h) g) 5 5 i) 6 Answers: 8 = 8 ( ) = f) ( / ) 6 = = g) 5 / 5 / =5 7/ h) ( / ) / = /6 i) ( ) /6 = /6 = / or Return to Review Topic

13 Math 08 T6 Eercise 5 Answers Page Evaluate the following. log log log 8 log 0 00 log 00 0 f) log 00 0 Answers: ; = ;/ = ;8/ = ; 0 = 00 ; 00/ =0 f) ; 00 / =0 = 0 Return to Review Topic