NAME: ALGEBRA 350 BLOCK 7. Simplifying Radicals Packet PART 1: ROOTS

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1 NAME: ALGEBRA 50 BLOCK 7 DATE: Simplifyig Radicals Packet PART 1: ROOTS READ: A square root of a umber b is a solutio of the equatio x = b. Every positive umber b has two square roots, deoted b ad b or collectively deoted as ± b. The positive square root, b, is called the pricipal square root. For example, the square roots of 16 are 16 = ad 16 = sice () = 16 ad ( ) = 16. The pricipal square root of 16 is. Sice squarig a umber will always result i a o-egative umber, the equatio x = b has o real solutios if b < 0. Therefore, a egative umber b has o real square roots. Examples: Solve. (a) x = 9 (b) x + = 0 (c) 5x = 15 x = ± 9 x = x = x = ± No real solutios x = ± x = or x = or DO: Solve. 1. x = 81. y 7 = 0. 16x = 5 READ: The iverse operatio from takig a square root is to square a umber. Squarig meas to multiply a umber by itself. Sice they are iverse operatios, if we square a square root, we cacel out the square root. For example: 5 = 5. Notice, though, that squarig a egative umber makes it positive: =. A cube root of a umber b is a solutio of the equatio x = b. Every umber b (positive, egative, or zero) has exactly oe real cube root, deoted b. The cube root of a positive umber is positive, ad the cube root of a egative umber is egative. For example: 8 = sice = 8, while 8 = sice ( ) = 8. (a) 6 (b) 10 6 (c) a 9 = = 10 sice (10 ) = 10 6 = a sice (a ) = a 9 = 100 1

2 We ca also take higher roots (fourth roots, fifth roots, etc.). Let s defie a geeral th root of b as a solutio of the equatio x = b. Notice: if =, we have the square root, ad if =, we have the cube root. From these two simple examples, we ca glea the followig geeral ideas: If is eve ad b > 0, there are two real th roots of b. o The pricipal (or positive) th root of b is deoted b. The other oe is b. If is eve ad b = 0, there is oe th root: 0 = 0. If is eve ad b < 0, there is o real th root of b. If is odd, there is exactly oe real th root of b, whether b is positive, egative, or zero. (a) 81 5 (b) 6 (c) 1 = = No real solutio We call ay umber ivolvig a root (square root, cube root, etc.) a radical. So, b DO: Simplify the followig radicals. is a radical for ay ad b ( 7) 8. a PART : PROPERTIES OF RADICALS READ: We kow 9 = 6 = 6. Ad 9 = = 6. So 9 = 9. This is true i geeral: ab a b = a = a b b NOTE: this is ot true if we are addig or subtractig udereath the radical = 5 = 5, but = + = 7, ot 5.

3 (a) 98 (b) 5 10 (c) 81 8 (d) 60 5 = 9 = 5 10 = 81 = = 9 = 50 = 7 = 15 = 15 = 5 = 7 = 7 = = 1 = = (e) a b (f) 6w (g) x5 y (h) a + b = a b = 6 w = x5 = (a + b ) y = a b = 6 w w = x x y = a + b = a b = 6w w = x x y READ: A radical expressio is ot cosidered simplified if there is a radical i the deomiator of a fractio. So we get rid of it via a process kow as ratioalizig the deomiator. It ivolves multiplyig the whole fractio by 1 (so that it does t chage the value of the fractio) but i a creative way. Here are examples of how it works: Examples: (a) 5 = 5 = 5 = 15 = 15 Radicals are i simplest radical form if: (b) x = x x x = x = x x x The umber udereath the radical has o factors that are perfect th powers (other tha 1). Every deomiator has bee ratioalized so that o radicals are i the deomiators of fractios. All of the aswers to the examples o this page are i simplest radical form. If your directios are to simplify, put all aswers i simplest radical form.

4 DO: Simplify x a 5 0. x y 1. 7x y. 16a + 16b. x + x +. 7x y 5. a b

5 PART : SUMS OF RADICALS READ: We ca oly add or subtract radicals if they are like terms. What makes two radicals like terms? They eed to have the same radicad (the umber udereath the square root). For example, ad 5 are like terms but 5 ad 7 are ot. They also eed to be the same type of root (both square roots or both cube roots, etc.). Oce you have like terms, you ca add them together by usig the distributive property. (a) (b) 81 (c) + = + 7 = = 9 = 1 = = + = 5 = 5 = 5 6 (d) 6 + (e) (f) 1x 5 x x + 5x x = = = x x x x x + 5x x = = = x x x x + 5x x = + = 6x x DO: Simplify

6 a b + 8a b PART : BINOMIALS CONTAINING RADICALS READ: We multiply two biomials cotaiig radicals by FOILig, just as we ormally would. (a) (b) 6 (c) = = 6 6 = = = = = = = = 18 1 = 6 NOTICE: i example (c), we got a whole umber. That is because the two biomials we multiplied together are idetical except for the sig i the middle. These biomials are called cojugates. Whe we multiply two cojugate biomials, the middle terms after we FOIL cacel out sice they are opposite sigs, leavig us with just whole umbers. To ratioalize a deomiator cotaiig a biomial, we multiply by the cojugate of the deomiator over itself. Example: Simplify = = = = DO: Simplify