, 1 1, A complex fraction is a quotient of rational expressions (including their sums) that result

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1 RT. Complex Fractions Wen working wit algebraic expressions, sometimes we come across needing to simplify expressions like tese: xx 9 xx +, xx + xx + xx, yy xx + xx + +, aa Simplifying Complex Fractions A complex fraction is a quotient of rational expressions (including sums of rational expressions) were at least one of tese expressions contains a fraction itself. In tis section, we will examine two metods of simplifying suc fractions. Definition. A complex fraction is a quotient of rational expressions (including teir sums) tat result in a fraction wit more tan two levels. For example, as tree levels wile xx as four xx levels. Suc fractions can be simplified to a single fraction wit only two levels. For example,, oooo xx xx xx xx xx Tere are two common metods of simplifying complex fractions. Metod I (multiplying by te reciprocal of te denominator) Replace te main division in te complex fraction wit a multiplication of te numerator fraction by te reciprocal of te denominator fraction. We ten simplify te resulting fraction if possible. Bot examples given in Definition. were simplified using tis strategy. Metod I is te most convenient to use wen bot te numerator and te denominator of a complex fraction consist of single fractions. However, if eiter te numerator or te denominator of a complex fraction contains addition or subtraction of fractions, it is usually easier to use te metod sown below. Metod II (multiplying by LCD) Multiply te numerator and denominator of a complex fraction by te least common denominator of all te fractions appearing in te numerator or in te denominator of te complex fraction. Ten, simplify te resulting fraction if possible. For example, to simplify yy+ xx xx+ yy, multiply te numerator yy + and te denominator xx + by te LLLLLL xx yy, xxxx. So, xx yy yy + xx xx + xxxx yy xxxx xxyy + yy yy(xxxx + ) xx yy + xx xx(xxxx + ) yy xx

2 Simplifying Complex Fractions Use a metod of your coice to simplify eac complex fraction. a. xx xx xx xx 5 xx +8xx+ xx 5xx aa + c. xx + 5 xx d. xx 5 xx+ xx xx a. Since te expression xx xx xx xx 5 xx +8xx+ xx 5xx denominator, we will simplify it using metod I, as below. contains a single fraction in bot te numerator and xx xx 8 xx xx 5 xx + 8xx + xx xx (xx )(xx + ) (xx )(xx + ) (xx )(xx ) (xx 5)(xx + ) (xx + )(xx + ) (xx 55)(xx + ) factor and multiply by te reciprocal aa + can be simplified in te following two ways: Metod I Metod II aa + +aa ()aa aa + aa ()aa ()(aa aa ) aa aa aa aa + aa ()aa aa +aa ()aa ()(aa aa ) aa aa aa Caution: In Metod II, te factor tat we multiply te complex fraction by must be equal to. Tis means tat te numerator and denominator of tis factor must be exactly te same. c. To simplify xx + 5 xx, we will use metod II. Multiplying te numerator and denominator by te LLLLLL, 5, we obtain 5 xx + 5 xx

3 8 d. Again, to simplify xx 5 xx+ xx xx, we will use metod II. Notice tat te lowest common multiple of te denominators in blue is (xx + )(xx ). So, after multiplying te numerator and denominator of te wole expression by te LCD, we obtain xx 5 xx + xx xx 5xx + xx (xx + )(xx ) 5(xx ) 5xx + 0 (xx + )(xx ) (xx + ) xx Simplifying Rational Expressions wit Negative Exponents Simplify eac expression. Leave te answer wit only positive exponents. a. xx yy yy xx aa aa a. If we write te expression wit no negative exponents, it becomes a complex fraction, wic can be simplified as in Example. So, xx yy yy xx xx yy yy xx xxxx xxxx yy xx xxxx(yy xx) xxxx As above, first, we rewrite te expression wit only positive exponents and ten simplify as any oter complex fraction. aa aa aa aa aa aa Remember! Tis factor must be aa aa aa ( aa) Simplifying te Difference Quotient for a Rational Function Find and simplify te expression ff(aa+) ff(aa) for te function ff(xx) xx+. Since ff(aa + ) aa++ and ff(aa) aa+, ten ff(aa + ) ff(aa) aa + + aa +

4 9 To simplify tis expression, we can multiply te numerator and denominator by te lowest common denominator, wic is (aa + + )(aa + ). Tus, aa + + aa + aa + aa (aa + + )(aa + ) keep te denominator in a factored form (aa + + )(aa + ) aa + (aa + + ) (aa + + )(aa + ) (aa + + )(aa + ) (aa + + )(aa + ) Tis bracket is essential! (aa + + )(aa + ) RT. Exercises Vocabulary Ceck Complete eac blank wit te most appropriate term from te given list: complex, LCD, multiply, reciprocal, same, single.. A quotient of two rational expressions tat results in a fraction wit more tan two levels is called a algebraic fraction.. To simplify a complex fraction means to find an equivalent fraction PP, were PP and QQ are QQ polynomials wit no essential common factors.. To simplify a complex rational expression using te metod, first write bot te numerator and te denominator as single fractions in simplified form.. To simplify a complex fraction using te metod, first find te LCD of all rational expressions witin te complex fraction. Ten, te numerator and denominator by te LCD expression. Concept Ceck Simplify eac complex fraction Simplify eac complex rational expression. 9. xx yy xx yy 0. nn 5 nn nn 5 8nn. aa + aa. nn + 5 nn. 9 xx xx + xx xx. 9 yy 5 yy 5. xx yy xx + yy. aa + aa

5 80. aa aa aa 8. xx yy xx yy xxxx 9. yy yy xx xx yy 0. 5 pp qq 5qq 5 pp. nn nn +nn nn +. tt tt + tt tt. aa aa. (xx+) xx 5. + xx xx. + xx+ + xx. aa aa 8. xx yy xx + yy 9. xx+ xx xx xx xx + xx 0. xx +xx+9 + xx+ xx 9 + xx. aa aa +. pp pp+9 pp +pp 5 pp 5pp+8 pp pp 0 Discussion Point. Are te expressions xx +yy xx +yy and xx +yy xx+yy equivalent? Explain wy or wy not. Simplify eac expression. Leave your answer wit only positive exponents.. aa 5. xx + xx xx. xx yy xx (nn+). + (nn+) Analytic Skills 8. ff(xx) 5 xx Find and simplify te difference quotient ff(aa+) ff(aa) 9. ff(xx) xx 0. ff(xx) xx for te given function.. ff(xx) xx Analytic Skills. aa aa aa aa Simplify eac continued fraction.. xx. aa + aa + aa