Radicals and Fractional Exponents


 Millicent Curtis
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1 Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it is askig you is what umber multiplied by its self umber of times will equal X? Ex. 6 Square Roots Cube Roots th Roots = because x= 8 = 6, because 6x6=6 7 =, because xx= 8 16 =, because xx=7 81 =, because xxx=16 =, because xxx=81 Perfect Roots Perfect roots are roots that ca be perfectly broke dow like i the examples above. Here is a list of the most commo perfect roots. These should be memorized! Perfect Square Roots 1 = 1 6 = 8 1 = 81 = = 100 = = 11 = 11 6 = 1 = = = = 7 Solvig Imperfect Radical Expressios Perfect Cube, Fourth, ad Fifth Roots 16 = 1 81 = 6 = 6 = 1 = = 1 Imperfect radical expressios are umbers that do ot have perfect roots. For example, there is o umber that whe multiplied by itself will give you, except a decimal. However, we still have to simplify them as much as we ca. The easiest way to do it is to break the umber dow ito a product of its primes by usig a factor tree. Oce that is doe, every umber that repeats itself umber of times ca be pulled out of the radical, everythig else remais iside. = = = = = 1 = = Step 1. Break dow ito products of primes 1 /\ 6 x /\ \ x x Ex. 1 Step. Look umber repeatig times N = so look for umber that repeats twice. x x x x Step. Pull out of Radical goes i frot of radical, ad is left udereath. Radicals ad Fractioal Expoets Provided by Tutorig Services 1 Reviewed August 01
2 If more tha oe umber ca be pulled out from the radical, the you multiply them o the outside. 18 x xxx8 / / xxxxx / / / / / xxxxxx Ex. 18 N=, so look for umber repeatig times. xxxxxx=18 Pull each group out ad put i frot of radical sig ad multiply. x Aother way of solvig imperfect radical expressios is to break the umber dow ito a product of perfect squares (this is why it is importat to have them memorized!). The you ca solve each perfect square idividually, for Ex. 88 = 6xx = 6 x x = 6xx =1 Step 1. Break dow ito a product of perfect squares / 9 Ex. 7 Step. Simplify perfect squares idividually, ad leave what ca t be broke dow further uder the radical. 7= 9 x x 7= x x Step. Multiply umbers o the outside of radical. 7= x x 6 16 / 8 7 Ex = 8 x 7 x = x x = 6 Radical expressios with variables Some radical expressios will also iclude variables, ex. 16a b. To simplify, treat the umbers as always. The variables ca be simplified by dividig ito the expoet of the variable. However may times it is evely divisible is how may you ca take out; leave the remaider uder the radical. For example, a 7. N=, ad goes ito 7 twice with oe left over, so the I take two a s out ad leave oe uder the radical, a a. Ex. a b 8 Provided by Tutorig Services Radicals ad Fractioal Expoets
3 Step 1. Break dow umber Step. Break dow the a Step. Break dow the b 8x xxxx, so oe o the outside, two iside goes ito oce with zero left over. So oe a o the outside, oe iside. Fial aswer ab b goes ito 8 twice with two left over. So two b s o the outside, ad two iside Addig ad Subtractig Radical Expressios Whe addig or subtractig radicals you treat them the same as you would a variable, you ca oly put like terms together. Both the idex, i.e. the value, ad what is uder the radical must be idetical i order to add or subtract. Just like a + a = a, 6 Multiplyig Radical Expressios + 6 Whe multiplyig radical expressio you simply eed to follow this rule, a = 6 6. x b = a x b. If there are coefficiets, you simply multiply them ormally. The fial step is to simplify if possible. Also, remember that i order to multiply, the idex must be the same, you caot multiply a square root with a cube root. Step 1. Multiply coefficiets ab x a = 8a b 9c 8a b 16b c Ex. ab 9c 8a b 6x7b c x a 18b Step. Multiply uder radical x 18b = 16b c Step. Simplify 8xa b 6b c Step. Ca you simplify? 8a b 16b c a b 6bc yes Dividig Radical Expressios Whe dividig radical expressios you eed to follow this rule, a = a. If there are b b coefficiets, you simply divide them ormally. The simplify what is uder the radical as much as possible, ad the simplify the radical itself if possible. Remember, i order to divide the degree must be the same for both radical expressios. Step 1. Rewrite as 1 Radical Ex. 16a 7 b a b Step. Simplify uder Radical 16a7 b a b a b Step. Simplify Radical a b a ab 7 a b Provided by Tutorig Services Radicals ad Fractioal Expoets
4 Expoets Expoets are very much like the reverse of roots. Rather tha what umber multiplied by itself umber of times equals X as with the radical X, X is askig X multipled by itself umber of times equals what? For example = 81 because xxx=81. Notice that 81 some rules ad properties for workig with expoets. =. Here are Addig ad Subtractig Multiplyig Dividig Must be same degree, oly add/subtract the coefficiets. Ex. x + x = x Power to power Add the expoets, a x a m = a (+m) Ex.6a x a = 18a Multiply the expoets for the variable, apply expoet to coefficiet. (a ) m = a xm Ex.. (a ) = a 1 = 81a 1 Subtract expoets, a / a m = a ( m) Ex. 6a /a = a Negative Expoets Move from umerator to deomiator or vice versa to make expoet positive. x = 1/ x Ex. ( ) = = = 7/1 Fractioal Expoets Fractioal Expoets must be simplified a differet way tha ormal expoets. For example, 1/. You caot multiply by its self ½ times. Sice Radicals ad expoets are reverses of each other, we ca switch from expoetial form to radical form to simplify. I order to do that, simply follow this formula: x /m m = x. Ex. 16 1/ = 16 = Ex. / = = 6 = 16 = Practice Problems (Simplify) 1. x y 7. 8x y 6. 81a 8 b 1. 6a 8 b 1. x 1x y 6. x 7 y 7. 16x y + x y x 18x y xy x 7y x xy 6 1. (x ) 1. (a b 9 ) / 8. (x 1/ y / ) 6 /(x y 8 ) ab / ab 10. 1ab a b / 1. (/9) / 1. a 1/ a / Provided by Tutorig Services Radicals ad Fractioal Expoets
5 Solutios 1. x y y. xy. a b. ab a b. 6x y 6. x y x 7. xy x 8. 6x /y 6 9. b 1b 10. ab a b /1 1. a 1. 1/x 1. a b 6 Provided by Tutorig Services Radicals ad Fractioal Expoets
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