Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Section 7.1: Radicals and Rational ExponentsDefinition of nth root of a numberLet a and b be real numbers and let n be an integer n ≥ 2. If a = bn, then b is an nth root of a. If n = 2, theroot is called square root. If n = 3, the root is called cube root.Definition of principal nth root of a numberLet a be a real number that has at least one (real number) nth root. The principal nth root of a is the nth rootthat has the same sign as a and it is denoted by the radical symboln√a. The positive integer n is the indexof the radical, and a is the radicand.Ex.1Examples of nth roots.(1) 3 =√9(2) −5 =√25(3) 2 =4√16(4) 4 =3√64Ex.2Find each principal root.(1)√36(2) −√36(3)√−4(4)3√8(5)3√−81Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Properties of nth rootsLet a be a real number.(1) If a is positive and n is even, then a has exactly two real nth roots, which are denoted byn√a and−n√a.(2) If n is odd (a is any real number), then a has one real nth root, which is denoted byn√a.(3) If a is negative and n is even, then a has no (real) nth root.Ex.3(1) 81 has two real square roots:√9 = 3 and −√9 = −3.(2)3√27 = 3(3)√−25 has no real square root.Perfect squares and perfect cubesA perfect square is an integer which is a square of an integer. A perfect cube is an integer which is a cube of aninteger.Ex.4State whether each number is a perfect square, a perfect cube, both, or neither.(1) 81(2) −125(3) 64(4) 32(5) 12Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Properties of nth powers and nth rootsLet a be a real number and n be an integer n ≥ 2.(1) If n is odd, then (n√a)n= a.(2) If n is even, then (n√a)n= |a|.Ex.5Evaluate each radical expression(1) (√5)2(2)p(−5)2(3)3√43(4)p(−3)2(5)p−(32)Definition of rational exponentsLet a be a real number and let n be an integer such that n ≥ 2. If the principal nth root of a exists, thena1n=n√aIf m is a positive integer that has no common factor with n, thenamn= (a1n)m= (n√a)mand amn= (am)1n=n√amRules of ExponentsLet m and n be rational numbers, and let a and b represent real numbers, variables, or algebraic expressions,a 6= 0, b 6= 0.(1) am· an= am+n(2)aman= am−n(3) (ab)m= am· bm(4) (am)n= amn(5)abm=ambm(6) a0= 1(7) a−m=1am(8)ab−m=bam3Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.6Evaluate each expression.(1) 843(2) 25−32(3) (64125)23(4) −1612(5) (−16)124Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.7Rewrite each expression using rational exponents.(1) x4√x3(2)3√x2√x3(3)3px2yEx.8Use the rule of exponents to simplify each expression.(1)p3√x(2)(2x−1)433√2x−15Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Definition of radical functionA radical function is a function that contains a radical.Ex.9Evaluate each radical function when x = 4.(1) f(x) =3√x − 31(2) g(x) =√16 − 3xDomain of a radical functionLet n be an integer, n ≥ 2.• If n is odd, the domain of f(x) =n√x is the set of all real numbers.• If n is even, the domain of f(x) =n√x is the set of all non-negative real numbers.Ex.10Describe the domain of each radical function.(1) f(x) =3√x(2) g(x) =√x36Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.11Find the domain off(x) =√2x − 1Section 7.2: Simplifying Radical ExpressionsProduct and Quotient Rules for RadicalsLet u and v be real numbers, variables, or algebraic expressions. If the nth roots of u and v are real, thefollowing rules are true.•n√uv =n√un√v•nruv=n√un√v, v 6= 0Ex.1Simplify each radical by removing as many factors as possible.(1)√12(2)√48(3)√75(4)√1627Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.2Simplify each radical expression.(1)√25x2(2)√12x3(3)√144x4(4)3√40(5)5√486x7(6)3p128x3y5(7)q8125(8)√56x2√88Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.3Simplify−3ry527x3Simplifying Radical ExpressionsA radical expression is in the simplest form if(1) All possible nth powered factors have been removed from each radical.(2) No radical contains a fraction.(3) No denominator of a fraction contains a radical.Ex.4Rationalize the denominator in each radical expression.(1)q35(2)43√9(3)83√189Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.5Rationalize the denominator in each radical expression.(1)q8x12y5(2)3q54x6y35z2Ex.6Find the length of the hypothenuse of the following right triangleEx.7A softball diamond has the shape of a square with 60-foot sides. The catcher is 5 feet behind home plate.How far does the catcher have to throw to reach second base?10Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Section 7.3: Adding and Subtracting Radical ExpressionsLike RadicalsTwo or more radical expressions are like radicals if they have the same index and the same radicand.Ex.1Simplify each radical expression by combining like radicals(1)√7 + 5√7 − 2√7(2) 33√x + 23√x +√x − 8√x(3)√45x + 3√20x(4) 53√x − x√4x(5)3√6x4+3√48x −3√162x411Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.2Simplify√7 −5√7Section 7.4: Multiplying and Dividing Radical ExpressionsEx.1Find each product and simplify(1)√6 ·√3(2) 33√5 ·3√16(3)√3(2 +√5)(4)√2(4 −√8)(5)√6(√12 −√3)12Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.2Find the product and simplify(1) (2√7 − 4)(√7 + 1)(2) (3 −√x)(1 +√x)Ex.3Find each conjugate of the expression and multiply the expression by its conjugate(1) 2 −√5(2)√3 +√x13Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.4Simplify(1)√31 −√5(2)42 −√3(3)5√2√7 +√214Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Ex.5Perform each division and simplify(1) 6 ÷ (√x − 2)(2) (2 −√3) ÷ (√6 +√2)(3) 1 ÷ (√x −√x + 1)15Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010Section 7.5: Radical Equations and ApplicationsRaising each side of an equation to the nth powerLet u and v be numbers, variables, or algebraic expressions, and let n be a positive integer. If u = v, then itfollows that un= vn. This is called raising each side of an equation to the nth power.Ex.1Solve√x − 8 = 0Ex.2Solve√3x + 6 = 016Chapter 7: Radicals and Complex Numbers Lecture notes Math
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