Sample Problems. 2. Rationalize the denominator in each of the following expressions Simplify each of the following expressions.

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1 Lecture Notes Radical Exressions age Samle Problems. Simlify each of the following exressions. Assume that a reresents a ositive number. a) b) g) c) h) 7 x y d) 0 + i) j) x x + e) k) ( x) ( + x) f) 0a 0a + a l) x. Rationalize the denominator in each of the following exressions. a) b) 0 c) 7 + d) e) +. Simlify each of the following exressions. a) b) c) + d) e) Find the exact value of x x + 6 if x.. Solve each of the following quadratic equations by comleting the square. Check your solution(s). a) x x + b) x + x Practice Problems. Simlify each of the following exressions. Assume that a reresents a ositive number. a) 0 e) + b) f) c) 0 + g) d) a 7 a + 0a h) 7 + c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

2 Lecture Notes Radical Exressions age. Rationalize the denominator in each of the following exressions. a) 7 c) 0 + e) + g) x + x + b) 7 d) 7 + f) h) x +. Find the exact value of a) x 6x + if x 0 b) b + b 0 if b c) x 0x + 6 if x 6 +. Solve each of the following quadratic equations by comleting the square. Check your solution(s). a) x + 7 x b) x + x + 0 c) x x + Samle Problems Answers. a) b) c) x y xy d) e) f) a a g) h) 7 i) j) x + x 6 k) 6 + x x l) x x +. a) b) + 0 c) + 7. a) 7 b) 7 c) d) e) d) + e) 0.. a) ; + b) ; + Practice Problems Answers. a) b) c) d) 7a a e) f) 9 g) 7 h) 0. a) 7 7 e) b) f) c) + 0 d) + 7 or + 7 g) x + h) x x 6. a) b) 7 + c). a) 7 ; 7 + b) 6 ; 6 + c) ; + c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

3 Lecture Notes Radical Exressions age Samle Problems Solutions. Simlify each of the following exressions. Assume that a reresents a ositive number. a) We rst factor the number under the square root sign into two factors, where the rst factor is the largest square we can nd in the number. Then this rst art can come out from under the square root sign and become a coe cient. 6 6 b) We rst factor the number under the square root sign into two factors, where the rst factor is the largest square we can nd in the number. Then this rst art can come out from under the square root sign and become a coe cient. 9 9 c) x y We rst factor the exressions under the square root sign into two factors, where the rst factor is the largest square we can nd in the exression. Then this rst art can come out from under the square root sign. x y 6x y xy 6x y xy x y xy d) 0 + We simlify each exression as in the revious roblem. Then we combine like radicals ( + ) e) We simlify both radical exressions and simlify c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

4 Lecture Notes Radical Exressions age f) 0a 0a + a 0a 0a + a 6a 0 a 6a 0 a + 9a 0 a 6a 0 a 6a 0 a + 9a 0 a a a 6a a + 7a a a a a a + a a ( + ) a a a a Note: We can rearrange the nal answer as a a a a. This other form is just as correct and might even be referable in some cases. g) h) i) We will rst work out and then multily the result by. + + j) x x + x x x x + x x x + x x 6 x + x 6 c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

5 Lecture Notes Radical Exressions age k) ( x) ( + x) x + x 6 + x x x x 6 + x x l) x x x x we FOIL x x x x + x x x + x x +. Rationalize the denominator in each of the following exressions. a) To rationalize the denominator, we will multily both the numerator and denominator by. b) 0 To rationalize the denominator, we will multily both the numerator and denominator by the conjugate of the denominator, which is The denominator is since c) 7 + To rationalize the denominator, we will multily both the numerator and denominator by the conjugate of the denominator, which is d) c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

6 Lecture Notes Radical Exressions age 6 e) + We multily the fraction by as a fraction of whose both numerator and denominator are the conjugate of the denominator We FOIL out both numerator and denominator Note: although this answer is accetable, the exression can be further simli ed Simlify each of the following exressions. a) b) We rst bring the fractions to the common denominator - they are conjugates, that hels, because the common denominator is then rational. We then multily the factors (7) c) + q + q q + q r + s 9 c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

7 Lecture Notes Radical Exressions age 7 d) q + q q q r s e) q 6 + q + 6 q 6 + q q q 6 + r Find the exact value of x x + 6 if x. We work out x rst. x Now we substitute x into x x + 6: x x Solve each of the following quadratic equations by comleting the square. Check your solution(s). a) x x + We factor by comleting the square. x x + {z } x x + reduce one side to zero x x 0 (x ) x x + 0 (x ) 0 (x ) 0 x + x 0 x and x + c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0

8 Lecture Notes Radical Exressions age We check: if x, then LHS RHS and if x +, then LHS RHS So, both solutions are correct. b) x + x We factor by comleting the square. x x + 6 {z } x x + 0 (x ) x x (x ) 0 (x ) 0 x + x 0 x and x + We check: if x, then LHS + RHS and if x +, then LHS RHS + + So, both solutions are correct. For more documents like this, visit our age at htt:// and click on Lecture Notes. questions or comments to mhidegkuti@ccc.edu. c coyright Hidegkuti, Powell, 00 Last revised: January 7, 0