14.1 Understanding Rational Exponents and Radicals

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1 Name Class Date 14.1 Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? Resource Locker Explore 1 Uderstadig Iteger Expoets Recall that powers like are evaluated by repeatig the base () as a factor a umber of times equal to the expoet (). So = = 9. What about a egative expoet, or a expoet of 0? You caot write a product with a egative umber of factors, but a patter emerges if you start from a positive expoet ad divide repeatedly by the base. Startig with powers of : = = 1 = Dividig a power of by is equivalet to the expoet by. Complete the patter: _ _ 1 _ 0 _ -1 _ - 7 _ 9 _ _ _ _ -1 =, - = 9 = Houghto Miffli Harcourt Publishig Compay Iteger expoets less tha 1 ca be summarized as follows: Words Numbers Variables Ay o-zero umber raised to the power of 0 is 1; 0 0 is udefied 0 = x = 1 for x 0 (.4) = 1 Ay o-zero umber raised to a egative power is equal to 1 divided - by the same umber raised to the opposite, positive power. = = 9 Reflect 1. Discussio Why does there eed to be a exceptio i the secod rule for the case of x = 0? - x = for x 0, x ad iteger. Module Lesso 1

2 Explore Explorig Ratioal Expoets A radical expressio is a expressio that cotais the radical symbol,. For a, is called the idex ad a is called the radicad. must be a iteger greater tha 1. a ca be ay real umber whe is odd, but must be o-egative whe is eve. Whe =, the radical is a square root ad the idex is usually ot show. You ca write a radical expressio as a power. First, ote what happes whe you raise a power to a power. ( ) = ( ) = ( ) ( ) = 6, so ( ) =. I fact, for all real umbers a ad all ratioal umbers m ad, ( a m ) = a m. This is called the Power of a Power Property. A radical expressio ca be writte as a expoetial expressio: a = a k. Fid the value for k whe =. Start with the equatio. a = a k Square both sides. ( a ) = ( a k ) Defiitio of square root = ( a k ) Power of a power property a 1 = a Reflect. What do you thik will be the rule for other values of the radical idex? Equate expoets. 1 = Solve for k. k = Explai 1 Simplifyig Numerical Expressios with th Roots For ay iteger > 1, the th root of a is a umber that, whe multiplied by itself times, is equal to a. x = a x = a The th root ca be writte as a radical with a idex of, or as a power with a expoet of _ 1. A expoet i the form of a fractio is a ratioal expoet. a = a The expressios are iterchageable, ad to evaluate the th root, it is ecessary to fid the umber, x, that satisfies the equatio x = a. Example 1 Fid the root ad simplify the expressio. 64 Covert to radical. 64 = 64 Rewrite radicad as a power. = Defiitio of th root = Covert to radicals = Rewrite radicads as powers. = Houghto Miffli Harcourt Publishig Compay Apply defiitio of th root. = + Simplify. = Module Lesso 1

3 Your Tur Explai Simplifyig Numerical Expressios with Ratioal Expoets Give that for a iteger greater tha 1, b = b, you ca use the Power of a Power Property to defie b m_ for ay positive iteger m. b m_ = b m m_ b = b m = ( b m ) = ( b ) m m Power of a Power Property = ( b ) Defiitio of b _ = b m The defiitio of a umber raised to the power of m is the th root of the umber raised to the mth power. The power of m ad the th root ca be evaluated i either order to obtai the same aswer, although it is geerally easier to fid the th root first whe workig without a calculator. Example Simplify expressios with fractioal expoets. 7 _ Defiitio of b m 7 = ( 7 ) Rewrite radicad as a power. = ( ) Defiitio of cube root = Houghto Miffli Harcourt Publishig Compay 5 _ = 9 Defiitio of b m 5 _ = ( 5 ) Rewrite radicad as a power. = ( ) Defiitio of root = 5 = Module Lesso 1

4 Your Tur 5. _ 5 5_ _ Elaborate 7. Why ca you evaluate a odd root for ay radicad, but eve roots require o-egative radicads? 8. I evaluatig powers with ratioal expoets with values like, why is it usually better to fid the root before the power? Would it chage the aswer to switch the order? 9. Essetial Questio Check-I How ca radicals ad ratioal expoets be used to simplify expressios ivolvig oe or the other? Houghto Miffli Harcourt Publishig Compay Module Lesso 1

5 Evaluate: Homework ad Practice Evaluate the expressios Olie Homework Hits ad Help Extra Practice ( ) 5. (-) Fid the root(s) ad simplify the expressio Houghto Miffli Harcourt Publishig Compay Module Lesso 1

6 Simplify the expressios with ratioal expoets _ _ Simplify the expressios _ Houghto Miffli Harcourt Publishig Compay Module Lesso 1

7 19. 5 _ ( 4 ) _ _ Geometry The volume of a cube is related to the area of a face by the formula V = A _. Houghto Miffli Harcourt Publishig Compay What is the volume of a cube whose face has a area of 100 cm? Module Lesso 1

8 4. Biology The approximate umber of Calories, C, that a aimal eeds each day is give by C = 7m _ 4, where m is the aimal s mass i kilograms. Fid the umber of Calories that a 16 kilogram dog eeds each day. 5. Rocket Sciece Escape velocity is a measure of how fast a object must be movig to escape the gravitatioal pull of a plaet or moo with o further thrust. The escape velocity for the moo is give approximately by the equatio V = 5600 ( d_ 1000 ) _ 1, where v is the escape velocity i miles per hour ad d is the distace from the ceter of the moo (i miles). If a luar lader thrusts upwards util it reaches a distace of 16,000 miles from the ceter of the moo, about how fast must it be goig to escape the moo s gravity? 6. Multiple Respose Which of the followig expressios caot be evaluated? a. 4 b. (-4) c. 4 d. (-4) e. 0 f. 0 Houghto Miffli Harcourt Publishig Compay WilleeCole/Alamy Module Lesso 1

9 H.O.T. Focus o Higher Order Thikig 7. Explai the Error Yua is asked to evaluate the expressio (-8) _ o his exam, ad writes that it is usolvable because you caot evaluate a egative umber to a eve fractioal power. Is he correct, ad if so, why? If he is ot correct, what is the correct aswer? 8. Commuicate Mathematical Ideas Show that the th root of a umber, a, ca be expressed with a expoet of _ 1 for ay positive iteger,. Houghto Miffli Harcourt Publishig Compay 9. Explai the Error Gretche thiks she has figured out how to evaluate the square root of a egative umber. Explai why her solutio is flawed. (-1) ( -1) = ( -1) = ( -1 ) 0 = 1 The she solves for ( -1) which is the same thig as _ -1. (-1) ( -1) = 1 ( -1) = (-1) = 1 = 1 But the square root of -1 caot be 1, sice 1 1 = 1, ot -1. What mistake did she make? Module Lesso 1

10 Lesso Performace Task Carbo-14 datig is used to determie the age of archeological artifacts of biological (plat or aimal) origi. Items that are dated usig carbo-14 iclude objects made from boe, wood, or plat fibers. This method works by measurig the fractio of carbo-14 remaiig i a object. The fractio of the origial carbo-14 remaiig ca be expressed by the fuctio, f = ( t_ 5700 ), where t is the legth of time sice the orgaism died. a. Fill i the followig table to see what fractio of the origial carbo-14 still remais after the passage of time. t t_ 5700 Fractio of Carbo-14 Remaiig ,400 17,100 b. The duratio of 5700 years is referred to as the half-life of carbo-14 because the amout of carbo-14 drops i half 5700 years after ay startig poit (ot just t = 0 years). Verify this property by comparig the amout of remaiig carbo-14 after 11,400 years ad 17,100 years. c. Write the correspodig expressio for the remaiig fractio of uraium-4, which has a half-life of about 80,000 years. Houghto Miffli Harcourt Publishig Compay Blaie Harrigto III/ Alamy Module Lesso 1