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== 7.3#h(0.6em)Simplify Complex Rational Expressions

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Simplify: #math.equation(block: false, alt: "the fraction the fraction 3 over 5 over the fraction 9 over 10 .")[$frac(frac(3, 5), frac(9, 10)) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 1 minus the fraction 1 over 3 over 4 squared plus 4 times 5 .")[$frac(1 − frac(1, 3), 4^(2) + 4 · 5) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 54")[$frac(1, 54)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 1 over 2 x plus the fraction 1 over 4 equals the fraction 1 over 8 .")[$frac(1, 2 x) + frac(1, 4) = frac(1, 8) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x equals minus 4")[$x = − 4$]
]
]


=== Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:

#math.equation(block: true, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8, the fraction the fraction x over 2 over the fraction x y over 6")[$frac(frac(3, 4), frac(5, 8)) &  &  &  &  & frac(frac(x, 2), frac(x y, 6))$]In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Complex Rational Expression]

A #strong[complex rational expression] is a rational expression in which the numerator and/or the denominator contains a rational expression.
]

Here are a few complex rational expressions:

#math.equation(block: true, alt: "the fraction the fraction 4 over y minus 3 over the fraction 8 over y squared minus 9 the fraction the fraction 1 over x plus the fraction 1 over y over the fraction x over y minus the fraction y over x the fraction the fraction 2 over x plus 6 over the fraction 4 over x minus 6 minus the fraction 4 over x squared minus 36")[$frac(frac(4, y − 3), frac(8, y^(2) − 9)) #h(3em) frac(frac(1, x) + frac(1, y), frac(x, y) − frac(y, x)) #h(3em) frac(frac(2, x + 6), frac(4, x − 6) − frac(4, x^(2) − 36))$]Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

We have already seen this complex rational expression earlier in this chapter.

#math.equation(block: true, alt: "the fraction the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 over the fraction 2 x squared minus 8 x plus 3 over x squared minus 5 x plus 6")[$frac(frac(6 x^(2) − 7 x + 2, 4 x − 8), frac(2 x^(2) − 8 x + 3, x^(2) − 5 x + 6))$]We noted that fraction bars tell us to divide, so rewrote it as the division problem:

#math.equation(block: true, alt: "open parenthesis the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 close parenthesis divided by open parenthesis the fraction 2 x squared minus 8 x plus 3 over x squared minus 5 x plus 6 close parenthesis .")[$( frac(6 x^(2) − 7 x + 2, 4 x − 8) ) ÷ ( frac(2 x^(2) − 8 x + 3, x^(2) − 5 x + 6) ) .$]Then, we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.

#examplebox("Example 1")[][
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 6 over x minus 4 over the fraction 3 over x squared minus 16 .")[$frac(frac(6, x − 4), frac(3, x^(2) − 16)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the fraction 6 over x minus 4 over the fraction 3 over x squared minus 16")[$#h(4em) frac(frac(6, x − 4), frac(3, x^(2) − 16))$]]),
  [Rewrite the complex fraction as division.], [#math.equation(block: false, alt: "the fraction 6 over x minus 4 divided by the fraction 3 over x squared minus 16")[$#h(4em) frac(6, x − 4) ÷ frac(3, x^(2) − 16)$]],
  [Rewrite as the product of first times the #linebreak() reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 6 over x minus 4 times the fraction x squared minus 16 over 3")[$#h(4em) frac(6, x − 4) · frac(x^(2) − 16, 3)$]],
  [Factor.], [#math.equation(block: false, alt: "the fraction 3 times 2 over x minus 4 times the fraction open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis over 3")[$#h(4em) frac(3 · 2, x − 4) · frac(( x − 4 ) ( x + 4 ), 3)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 3 times 2 open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis over 3 open parenthesis x minus 4 close parenthesis")[$#h(4em) frac(3 · 2 ( x − 4 ) ( x + 4 ), 3 ( x − 4 ))$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 3 times 2 open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis over 3 open parenthesis x minus 4 close parenthesis")[$#h(4em) frac(cancel(3) · 2 cancel(( x − 4 )) ( x + 4 ), cancel(3) cancel(( x − 4 )))$]],
  [Simplify.], [#math.equation(block: false, alt: "2 open parenthesis x plus 4 close parenthesis")[$#h(4em) 2 ( x + 4 )$]],
))

Are there any value(s) of #emph[x] that should not be allowed? The original complex rational expression had denominators of #math.equation(block: false, alt: "x minus 4")[$x − 4$] and #math.equation(block: false, alt: "x squared minus 16 .")[$x^(2) − 16 .$] This expression would be undefined if #math.equation(block: false, alt: "x equals 4")[$x = 4$] or #math.equation(block: false, alt: "x equals −4 .")[$x = −4 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 2 over x squared minus 1 over the fraction 3 over x plus 1 .")[$frac(frac(2, x^(2) − 1), frac(3, x + 1)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 3 open parenthesis x minus 1 close parenthesis")[$frac(2, 3 ( x − 1 ))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 1 over x squared minus 7 x plus 12 over the fraction 2 over x minus 4 .")[$frac(frac(1, x^(2) − 7 x + 12), frac(2, x − 4)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis x minus 3 close parenthesis")[$frac(1, 2 ( x − 3 ))$]
]
]

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

#examplebox("Example 2")[][
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 1 over 3 plus the fraction 1 over 6 over the fraction 1 over 2 minus the fraction 1 over 3 .")[$frac(frac(1, 3) + frac(1, 6), frac(1, 2) − frac(1, 3)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A fraction with the sum of 1/3 and 1/6 in the numerator, and the difference of 1/2 and 1/3 in the denominator, representing a complex mathematical expression.], alt: "A fraction with the sum of 1/3 and 1/6 in the numerator, and the difference of 1/2 and 1/3 in the denominator, representing a complex mathematical expression.", caption: none)]),
  [Simplify the numerator and denominator. #linebreak() Find the LCD and add the fractions in the numerator. #linebreak() Find the LCD and subtract the fractions in the #linebreak() denominator.], [#figure(figph[A complex fraction illustrating the addition of 1/3 and 1/6 in the numerator, and the subtraction of 1/3 from 1/2 in the denominator, with common denominator steps shown.], alt: "A complex fraction illustrating the addition of 1/3 and 1/6 in the numerator, and the subtraction of 1/3 from 1/2 in the denominator, with common denominator steps shown.", caption: none)],
  [Simplify the numerator and denominator.], [#figure(figph[A fraction where the numerator is 2/6 + 1/6 and the denominator is 3/6 - 2/6, demonstrating addition and subtraction of fractions with a common denominator.], alt: "A fraction where the numerator is 2/6 + 1/6 and the denominator is 3/6 - 2/6, demonstrating addition and subtraction of fractions with a common denominator.", caption: none)],
  [Rewrite the complex rational expression as a division #linebreak() problem.], [#figure(figph[A mathematical expression showing the division of two fractions: 3/6 ÷ 1/6.], alt: "A mathematical expression showing the division of two fractions: 3/6 ÷ 1/6.", caption: none)],
  [Multiply the first by the reciprocal of the second.], [#figure(figph[A mathematical expression showing the multiplication of two fractions: 3/6 multiplied by 6/1. The 'dot' symbol is used to denote multiplication between the fractions.], alt: "A mathematical expression showing the multiplication of two fractions: 3/6 multiplied by 6/1. The 'dot' symbol is used to denote multiplication between the fractions.", caption: none)],
  [Simplify.], [3],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 1 over 2 plus the fraction 2 over 3 over the fraction 5 over 6 plus the fraction 1 over 12 .")[$frac(frac(1, 2) + frac(2, 3), frac(5, 6) + frac(1, 12)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 14 over 11")[$frac(14, 11)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 3 over 4 minus the fraction 1 over 3 over the fraction 1 over 8 plus the fraction 5 over 6 .")[$frac(frac(3, 4) − frac(1, 3), frac(1, 8) + frac(5, 6)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 10 over 23")[$frac(10, 23)$]
]
]

We follow the same procedure when the complex rational expression contains variables.

#examplebox("Example 3")[How to Simplify a Complex Rational Expression using Division][
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 1 over x plus the fraction 1 over y over the fraction x over y minus the fraction y over x .")[$frac(frac(1, x) + frac(1, y), frac(x, y) − frac(y, x)) .$]

#solutionbox[
#figure(figph[Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.], alt: "Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.", caption: none)

#figure(figph[Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.], alt: "Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.", caption: none)

#figure(figph[Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.], alt: "Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 1 over x plus the fraction 1 over y over the fraction 1 over x minus the fraction 1 over y .")[$frac(frac(1, x) + frac(1, y), frac(1, x) − frac(1, y)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction y plus x over y minus x")[$frac(y + x, y − x)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction the fraction 1 over a plus the fraction 1 over b over the fraction 1 over a squared minus the fraction 1 over b squared")[$frac(frac(1, a) + frac(1, b), frac(1, a^(2)) − frac(1, b^(2)))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction a b over b minus a")[$frac(a b, b − a)$]
]
]

We summarize the steps here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify a complex rational expression by writing it as division.]

+ Simplify the numerator and denominator.
+ Rewrite the complex rational expression as a division problem.
+ Divide the expressions.
]

#examplebox("Example 4")[][
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction n minus the fraction 4 n over n plus 5 over the fraction 1 over n plus 5 plus the fraction 1 over n minus 5 .")[$frac(n − frac(4 n, n + 5), frac(1, n + 5) + frac(1, n − 5)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A complex fraction with a numerator of n minus 4n over n plus 5, and a denominator of 1 over n plus 5 plus 1 over n minus 5.], alt: "A complex fraction with a numerator of n minus 4n over n plus 5, and a denominator of 1 over n plus 5 plus 1 over n minus 5.", caption: none)]),
  [Simplify the numerator and denominator. #linebreak() Find common denominators for the numerator and #linebreak() denominator.], [#figure(figph[An algebraic expression showing a complex fraction. The numerator involves subtraction of two fractions, and the denominator involves addition of two fractions. Variables 'n' are present.], alt: "An algebraic expression showing a complex fraction. The numerator involves subtraction of two fractions, and the denominator involves addition of two fractions. Variables 'n' are present.", caption: none)],
  [Simplify the numerators.], [#figure(figph[A large algebraic fraction. The numerator is the difference of (n^2+5n)/(n+5) and 4n/(n+5). The denominator is the sum of (n-5)/((n+5)(n-5)) and (n+5)/((n-5)(n+5)).], alt: "A large algebraic fraction. The numerator is the difference of (n^2+5n)/(n+5) and 4n/(n+5). The denominator is the sum of (n-5)/((n+5)(n-5)) and (n+5)/((n-5)(n+5)).", caption: none)],
  [Subtract the rational expressions in the numerator and #linebreak() add in the denominator.], [#figure(figph[A mathematical expression showing a complex algebraic fraction. The numerator is (n^2 + 5n - 4n) / (n + 5) and the denominator is (n - 5 + n + 5) / ((n + 5)(n - 5)).], alt: "A mathematical expression showing a complex algebraic fraction. The numerator is (n^2 + 5n - 4n) / (n + 5) and the denominator is (n - 5 + n + 5) / ((n + 5)(n - 5)).", caption: none)],
  [Simplify. (We now have one rational expression over #linebreak() one rational expression.)], [#figure(figph[A complex algebraic fraction is displayed, with the numerator being (n^2 + n)/(n + 5) and the denominator being 2n/((n + 5)(n - 5)).], alt: "A complex algebraic fraction is displayed, with the numerator being (n^2 + n)/(n + 5) and the denominator being 2n/((n + 5)(n - 5)).", caption: none)],
  [Rewrite as fraction division.], [#figure(figph[A mathematical expression showing the division of two algebraic fractions: (n^2 + n) / (n + 5) divided by (2n) / ((n + 5)(n - 5)).], alt: "A mathematical expression showing the division of two algebraic fractions: (n^2 + n) / (n + 5) divided by (2n) / ((n + 5)(n - 5)).", caption: none)],
  [Multiply the first times the reciprocal of the second.], [#figure(figph[A mathematical expression showing the product of two fractions: (n^2 + n) / (n + 5) multiplied by (n + 5)(n - 5) / (2n).], alt: "A mathematical expression showing the product of two fractions: (n^2 + n) / (n + 5) multiplied by (n + 5)(n - 5) / (2n).", caption: none)],
  [Factor any expressions if possible.], [#figure(figph[A mathematical expression showing a fraction with n(n+1)(n+5)(n-5) in the numerator and (n+5)2n in the denominator, set against a plain white background.], alt: "A mathematical expression showing a fraction with n(n+1)(n+5)(n-5) in the numerator and (n+5)2n in the denominator, set against a plain white background.", caption: none)],
  [Remove common factors.], [#figure(figph[A fraction with algebraic terms in the numerator and denominator, showing cancellation of common factors 'n', '(n+5)', and '2' (implicitly from '2n').], alt: "A fraction with algebraic terms in the numerator and denominator, showing cancellation of common factors 'n', '(n+5)', and '2' (implicitly from '2n').", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction (n+1)(n-5) all divided by 2.], alt: "A mathematical expression showing the fraction (n+1)(n-5) all divided by 2.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction b minus the fraction 3 b over b plus 5 over the fraction 2 over b plus 5 plus the fraction 1 over b minus 5 .")[$frac(b − frac(3 b, b + 5), frac(2, b + 5) + frac(1, b − 5)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction b open parenthesis b plus 2 close parenthesis open parenthesis b minus 5 close parenthesis over 3 b minus 5")[$frac(b ( b + 2 ) ( b − 5 ), 3 b − 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by writing it as division: #math.equation(block: false, alt: "the fraction 1 minus the fraction 3 over c plus 4 over the fraction 1 over c plus 4 plus the fraction c over 3 .")[$frac(1 − frac(3, c + 4), frac(1, c + 4) + frac(c, 3)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over c plus 3")[$frac(3, c + 3)$]
]
]


=== Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in . We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by #math.equation(block: false, alt: "the fraction LCD over LCD")[$frac("LCD", "LCD")$] we are multiplying by 1, so the value stays the same.

#examplebox("Example 5")[][
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 1 over 3 plus the fraction 1 over 6 over the fraction 1 over 2 minus the fraction 1 over 3 .")[$frac(frac(1, 3) + frac(1, 6), frac(1, 2) − frac(1, 3)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A complex fraction with (1/3 + 1/6) in the numerator and (1/2 - 1/3) in the denominator, illustrating basic fraction arithmetic.], alt: "A complex fraction with (1/3 + 1/6) in the numerator and (1/2 - 1/3) in the denominator, illustrating basic fraction arithmetic.", caption: none)]),
  [The LCD of all the fractions in the whole expression is 6.], [],
  [Clear the fractions by multiplying the numerator and #linebreak() denominator by that LCD.], [#figure(figph[A mathematical expression presented as a fraction. The numerator is 6 multiplied by the sum of 1/3 and 1/6. The denominator is 6 multiplied by the difference of 1/2 and 1/3. The '6's are red.], alt: "A mathematical expression presented as a fraction. The numerator is 6 multiplied by the sum of 1/3 and 1/6. The denominator is 6 multiplied by the difference of 1/2 and 1/3. The '6's are red.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical expression showing a fraction where the numerator is (6 \* 1/3) + (6 \* 1/6) and the denominator is (6 \* 1/2) - (6 \* 1/3). The number 6 is highlighted in red in each term.], alt: "A mathematical expression showing a fraction where the numerator is (6 * 1/3) + (6 * 1/6) and the denominator is (6 * 1/2) - (6 * 1/3). The number 6 is highlighted in red in each term.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing a fraction with 2+1 in the numerator and 3-2 in the denominator.], alt: "A mathematical expression showing a fraction with 2+1 in the numerator and 3-2 in the denominator.", caption: none)],
  [], [#figure(figph[The fraction 3 over 1 is displayed in black text on a white background.], alt: "The fraction 3 over 1 is displayed in black text on a white background.", caption: none)],
  [], [#figure(figph[The number '3' is prominently displayed in a bold, black font against a stark white background.], alt: "The number '3' is prominently displayed in a bold, black font against a stark white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 1 over 2 plus the fraction 1 over 5 over the fraction 1 over 10 plus the fraction 1 over 5 .")[$frac(frac(1, 2) + frac(1, 5), frac(1, 10) + frac(1, 5)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 3")[$frac(7, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 1 over 4 plus the fraction 3 over 8 over the fraction 1 over 2 minus the fraction 5 over 16 .")[$frac(frac(1, 4) + frac(3, 8), frac(1, 2) − frac(5, 16)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 10 over 3")[$frac(10, 3)$]
]
]

We will use the same example as in . Decide which method works better for you.

#examplebox("Example 6")[How to Simplify a Complex Rational Expressing using the LCD][
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 1 over x plus the fraction 1 over y over the fraction x over y minus the fraction y over x .")[$frac(frac(1, x) + frac(1, y), frac(x, y) − frac(y, x)) .$]

#solutionbox[
#figure(figph[Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.], alt: "Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.", caption: none)

#figure(figph[Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.], alt: "Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.", caption: none)

#figure(figph[Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.], alt: "Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 1 over a plus the fraction 1 over b over the fraction a over b plus the fraction b over a .")[$frac(frac(1, a) + frac(1, b), frac(a, b) + frac(b, a)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction b plus a over a squared plus b squared")[$frac(b + a, a^(2) + b^(2))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 1 over x squared minus the fraction 1 over y squared over the fraction 1 over x plus the fraction 1 over y .")[$frac(frac(1, x^(2)) − frac(1, y^(2)), frac(1, x) + frac(1, y)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction y minus x over x y")[$frac(y − x, x y)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify a complex rational expression by using the LCD.]

+ Find the LCD of all fractions in the complex rational expression.
+ Multiply the numerator and denominator by the LCD.
+ Simplify the expression.
]

Be sure to start by factoring all the denominators so you can find the LCD.

#examplebox("Example 7")[][
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 2 over x plus 6 over the fraction 4 over x minus 6 minus the fraction 4 over x squared minus 36 .")[$frac(frac(2, x + 6), frac(4, x − 6) − frac(4, x^(2) − 36)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A complex fraction with 2/(x+6) in the numerator, and in the denominator, 4/(x-6) minus 4/(x^2-36).], alt: "A complex fraction with 2/(x+6) in the numerator, and in the denominator, 4/(x-6) minus 4/(x^2-36).", caption: none)]),
  [Find the LCD of all fractions in the complex rational #linebreak() expression. The LCD is #math.equation(block: false, alt: "x squared minus 36 equals open parenthesis x plus 6 close parenthesis open parenthesis x minus 6 close parenthesis")[$x^(2) − 36 = ( x + 6 ) ( x − 6 )$].], [],
  [Multiply the numerator and denominator by the LCD.], [#figure(figph[An algebraic fraction where both numerator and denominator are multiplied by (x+6)(x-6) as a step to simplify the expression and eliminate nested fractions.], alt: "An algebraic fraction where both numerator and denominator are multiplied by (x+6)(x-6) as a step to simplify the expression and eliminate nested fractions.", caption: none)],
  [Simplify the expression.], [],
  [Distribute in the denominator.], [#figure(figph[A mathematical expression displaying a fraction. The numerator has (x+6)(x-6) times 2/(x+6). The denominator subtracts two terms, both starting with (x+6)(x-6).], alt: "A mathematical expression displaying a fraction. The numerator has (x+6)(x-6) times 2/(x+6). The denominator subtracts two terms, both starting with (x+6)(x-6).", caption: none)],
  [Simplify.], [#figure(figph[A complex algebraic fraction showing several terms being canceled out, including (x+6) and (x-6), across the numerator and the two subtracted terms in the denominator.], alt: "A complex algebraic fraction showing several terms being canceled out, including (x+6) and (x-6), across the numerator and the two subtracted terms in the denominator.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction 2(x - 6) divided by 4(x + 6) - 4.], alt: "A mathematical expression showing the fraction 2(x - 6) divided by 4(x + 6) - 4.", caption: none)],
  [To simplify the denominator, distribute #linebreak() and combine like terms.], [#figure(figph[A mathematical expression showing a fraction with 2(x - 6) in the numerator and 4x + 20 in the denominator.], alt: "A mathematical expression showing a fraction with 2(x - 6) in the numerator and 4x + 20 in the denominator.", caption: none)],
  [Factor the denominator.], [#figure(figph[A mathematical fraction with 2(x-6) in the numerator and 4(x+5) in the denominator.], alt: "A mathematical fraction with 2(x-6) in the numerator and 4(x+5) in the denominator.", caption: none)],
  [Remove common factors.], [#figure(figph[A mathematical fraction showing the simplification of 2(x-6) over 2\*2(x+5), with one '2' in the numerator and one in the denominator crossed out.], alt: "A mathematical fraction showing the simplification of 2(x-6) over 2*2(x+5), with one '2' in the numerator and one in the denominator crossed out.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression displaying a fraction with (x - 6) in the numerator and 2(x + 5) in the denominator.], alt: "A mathematical expression displaying a fraction with (x - 6) in the numerator and 2(x + 5) in the denominator.", caption: none)],
  [Notice that there are no more factors #linebreak() common to the numerator and denominator.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 3 over x plus 2 over the fraction 5 over x minus 2 minus the fraction 3 over x squared minus 4 .")[$frac(frac(3, x + 2), frac(5, x − 2) − frac(3, x^(2) − 4)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 open parenthesis x minus 2 close parenthesis over 5 x plus 7")[$frac(3 ( x − 2 ), 5 x + 7)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 2 over x minus 7 minus the fraction 1 over x plus 7 over the fraction 6 over x plus 7 minus the fraction 1 over x squared minus 49 .")[$frac(frac(2, x − 7) − frac(1, x + 7), frac(6, x + 7) − frac(1, x^(2) − 49)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x plus 21 over 6 x minus 43")[$frac(x + 21, 6 x − 43)$]
]
]

Be sure to factor the denominators first. Proceed carefully as the math can get messy!

#examplebox("Example 8")[][
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 4 over m squared minus 7 m plus 12 over the fraction 3 over m minus 3 minus the fraction 2 over m minus 4 .")[$frac(frac(4, m^(2) − 7 m + 12), frac(3, m − 3) − frac(2, m − 4)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A complex algebraic fraction with 4 over the quadratic m^2 - 7m + 12 in the numerator, and the difference of two rational expressions, 3/(m-3) - 2/(m-4), in the denominator.], alt: "A complex algebraic fraction with 4 over the quadratic m^2 - 7m + 12 in the numerator, and the difference of two rational expressions, 3/(m-3) - 2/(m-4), in the denominator.", caption: none)]),
  [Find the LCD of all fractions in the #linebreak() complex rational expression.], [],
  [The LCD is #math.equation(block: false, alt: "open parenthesis m minus 3 close parenthesis open parenthesis m minus 4 close parenthesis .")[$( m − 3 ) ( m − 4 ) .$]], [],
  [Multiply the numerator and #linebreak() denominator by the LCD.], [#figure(figph[Algebraic expression demonstrating the simplification of a complex fraction by multiplying both the numerator and denominator by (m-3)(m-4).], alt: "Algebraic expression demonstrating the simplification of a complex fraction by multiplying both the numerator and denominator by (m-3)(m-4).", caption: none)],
  [Simplify.], [#figure(figph[A complex algebraic fraction showing cancellation of terms (m-3) and (m-4) in both the numerator and denominator, simplifying to 4 over (m-4)3 - (m-3)2.], alt: "A complex algebraic fraction showing cancellation of terms (m-3) and (m-4) in both the numerator and denominator, simplifying to 4 over (m-4)3 - (m-3)2.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction 4 divided by the quantity 3 times (m minus 4) minus 2 times (m minus 3).], alt: "A mathematical expression showing the fraction 4 divided by the quantity 3 times (m minus 4) minus 2 times (m minus 3).", caption: none)],
  [Distribute.], [#figure(figph[A mathematical expression showing the fraction 4 over the quantity 3m minus 12 minus 2m plus 6, which simplifies to 4 over m minus 6.], alt: "A mathematical expression showing the fraction 4 over the quantity 3m minus 12 minus 2m plus 6, which simplifies to 4 over m minus 6.", caption: none)],
  [Combine like terms.], [#figure(figph[A fraction is displayed with a numerator of 4 and a denominator of m minus 6.], alt: "A fraction is displayed with a numerator of 4 and a denominator of m minus 6.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 3 over x squared plus 7 x plus 10 over the fraction 4 over x plus 2 plus the fraction 1 over x plus 5 .")[$frac(frac(3, x^(2) + 7 x + 10), frac(4, x + 2) + frac(1, x + 5)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 5 x plus 22")[$frac(3, 5 x + 22)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction 4 y over y plus 5 plus the fraction 2 over y plus 6 over the fraction 3 y over y squared plus 11 y plus 30 .")[$frac(frac(4 y, y + 5) + frac(2, y + 6), frac(3 y, y^(2) + 11 y + 30)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 open parenthesis 2 y squared plus 13 y plus 5 close parenthesis over 3 y")[$frac(2 ( 2 y^(2) + 13 y + 5 ), 3 y)$]
]
]

#examplebox("Example 9")[][
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction y over y plus 1 over 1 plus the fraction 1 over y minus 1 .")[$frac(frac(y, y + 1), 1 + frac(1, y − 1)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A complex fraction with y/(y+1) in the numerator and 1 + 1/(y-1) in the denominator, representing a rational algebraic expression.], alt: "A complex fraction with y/(y+1) in the numerator and 1 + 1/(y-1) in the denominator, representing a rational algebraic expression.", caption: none)]),
  [Find the LCD of all fractions in the complex rational expression.], [],
  [The LCD is #math.equation(block: false, alt: "open parenthesis y plus 1 close parenthesis open parenthesis y minus 1 close parenthesis .")[$( y + 1 ) ( y − 1 ) .$]], [],
  [Multiply the numerator and denominator by the LCD.], [#figure(figph[An algebraic complex fraction. The numerator is (y+1)(y-1)y/(y+1) and the denominator is (y+1)(y-1)(1 + 1/(y-1)). The factor (y+1)(y-1) is shown in red.], alt: "An algebraic complex fraction. The numerator is (y+1)(y-1)y/(y+1) and the denominator is (y+1)(y-1)(1 + 1/(y-1)). The factor (y+1)(y-1) is shown in red.", caption: none)],
  [Distribute in the denominator and simplify.], [#figure(figph[Simplifying a rational algebraic expression by cancelling common factors (y+1) and (y-1) from the numerator and denominator of the fraction.], alt: "Simplifying a rational algebraic expression by cancelling common factors (y+1) and (y-1) from the numerator and denominator of the fraction.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical fraction with (y-1)y in the numerator and (y+1)(y-1) + (y+1) in the denominator.], alt: "A mathematical fraction with (y-1)y in the numerator and (y+1)(y-1) + (y+1) in the denominator.", caption: none)],
  [Simplify the denominator and leave the #linebreak() numerator factored.], [#figure(figph[A mathematical expression displaying a fraction. The numerator is y(y-1) and the denominator is y^2 - 1 + y + 1.], alt: "A mathematical expression displaying a fraction. The numerator is y(y-1) and the denominator is y^2 - 1 + y + 1.", caption: none)],
  [], [#figure(figph[A mathematical expression showing the fraction y(y-1) over y^2 + y.], alt: "A mathematical expression showing the fraction y(y-1) over y^2 + y.", caption: none)],
  [Factor the denominator and remove factors #linebreak() common with the numerator.], [#figure(figph[A mathematical fraction is shown, with y(y-1) as the numerator and y(y+1) as the denominator.], alt: "A mathematical fraction is shown, with y(y-1) as the numerator and y(y+1) as the denominator.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction (y - 1) / (y + 1). The numerator is 'y minus 1' and the denominator is 'y plus 1', separated by a horizontal fraction bar.], alt: "A mathematical expression showing the fraction (y - 1) / (y + 1). The numerator is 'y minus 1' and the denominator is 'y plus 1', separated by a horizontal fraction bar.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction the fraction x over x plus 3 over 1 plus the fraction 1 over x plus 3 .")[$frac(frac(x, x + 3), 1 + frac(1, x + 3)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x over x plus 4")[$frac(x, x + 4)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the complex rational expression by using the LCD: #math.equation(block: false, alt: "the fraction 1 plus the fraction 1 over x minus 1 over the fraction 3 over x plus 1 .")[$frac(1 + frac(1, x − 1), frac(3, x + 1)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x open parenthesis x plus 1 close parenthesis over 3 open parenthesis x minus 1 close parenthesis")[$frac(x ( x + 1 ), 3 ( x − 1 ))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access this online resource for additional instruction and practice with complex fractions.

- #link("https://openstax.org/l/37CompFrac")[Complex Fractions]
]


=== Key Concepts

- #strong[How to simplify a complex rational expression by writing it as division.]
+ Simplify the numerator and denominator.
+ Rewrite the complex rational expression as a division problem.
+ Divide the expressions.
- #strong[How to simplify a complex rational expression by using the LCD.]
+ Find the LCD of all fractions in the complex rational expression.
+ Multiply the numerator and denominator by the LCD.
+ Simplify the expression.


==== Practice Makes Perfect

#strong[Simplify a Complex Rational Expression by Writing it as Division]

In the following exercises, simplify each complex rational expression by writing it as division.

#math.equation(block: true, alt: "the fraction the fraction 2 a over a plus 4 over the fraction 4 a squared over a squared minus 16")[$frac(frac(2 a, a + 4), frac(4 a^(2), a^(2) − 16))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction a minus 4 over 2 a")[$frac(a − 4, 2 a)$]
]

#math.equation(block: true, alt: "the fraction the fraction 3 b over b minus 5 over the fraction b squared over b squared minus 25")[$frac(frac(3 b, b − 5), frac(b^(2), b^(2) − 25))$]

#math.equation(block: true, alt: "the fraction the fraction 5 over c squared plus 5 c minus 14 over the fraction 10 over c plus 7")[$frac(frac(5, c^(2) + 5 c − 14), frac(10, c + 7))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis c minus 2 close parenthesis")[$frac(1, 2 ( c − 2 ))$]
]

#math.equation(block: true, alt: "the fraction the fraction 8 over d squared plus 9 d plus 18 over the fraction 12 over d plus 6")[$frac(frac(8, d^(2) + 9 d + 18), frac(12, d + 6))$]

#math.equation(block: true, alt: "the fraction the fraction 1 over 2 plus the fraction 5 over 6 over the fraction 2 over 3 plus the fraction 7 over 9")[$frac(frac(1, 2) + frac(5, 6), frac(2, 3) + frac(7, 9))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 12 over 13")[$frac(12, 13)$]
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 2 plus the fraction 3 over 4 over the fraction 3 over 5 plus the fraction 7 over 10")[$frac(frac(1, 2) + frac(3, 4), frac(3, 5) + frac(7, 10))$]

#math.equation(block: true, alt: "the fraction the fraction 2 over 3 minus the fraction 1 over 9 over the fraction 3 over 4 plus the fraction 5 over 6")[$frac(frac(2, 3) − frac(1, 9), frac(3, 4) + frac(5, 6))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 20 over 57")[$frac(20, 57)$]
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 2 minus the fraction 1 over 6 over the fraction 2 over 3 plus the fraction 3 over 4")[$frac(frac(1, 2) − frac(1, 6), frac(2, 3) + frac(3, 4))$]

#math.equation(block: true, alt: "the fraction the fraction n over m plus the fraction 1 over n over the fraction 1 over n minus the fraction n over m")[$frac(frac(n, m) + frac(1, n), frac(1, n) − frac(n, m))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n squared plus m over m minus n squared")[$frac(n^(2) + m, m − n^(2))$]
]

#math.equation(block: true, alt: "the fraction the fraction 1 over p plus the fraction p over q over the fraction q over p minus the fraction 1 over q")[$frac(frac(1, p) + frac(p, q), frac(q, p) − frac(1, q))$]

#math.equation(block: true, alt: "the fraction the fraction 1 over r plus the fraction 1 over t over the fraction 1 over r squared minus the fraction 1 over t squared")[$frac(frac(1, r) + frac(1, t), frac(1, r^(2)) − frac(1, t^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction r t over t minus r")[$frac(r t, t − r)$]
]

#math.equation(block: true, alt: "the fraction the fraction 2 over v plus the fraction 2 over w over the fraction 1 over v squared minus the fraction 1 over w squared")[$frac(frac(2, v) + frac(2, w), frac(1, v^(2)) − frac(1, w^(2)))$]

#math.equation(block: true, alt: "the fraction x minus the fraction 2 x over x plus 3 over the fraction 1 over x plus 3 plus the fraction 1 over x minus 3")[$frac(x − frac(2 x, x + 3), frac(1, x + 3) + frac(1, x − 3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis over 2")[$frac(( x + 1 ) ( x − 3 ), 2)$]
]

#math.equation(block: true, alt: "the fraction y minus the fraction 2 y over y minus 4 over the fraction 2 over y minus 4 plus the fraction 2 over y plus 4")[$frac(y − frac(2 y, y − 4), frac(2, y − 4) + frac(2, y + 4))$]

#math.equation(block: true, alt: "the fraction 2 minus the fraction 2 over a plus 3 over the fraction 1 over a plus 3 plus the fraction a over 2")[$frac(2 − frac(2, a + 3), frac(1, a + 3) + frac(a, 2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over a plus 1")[$frac(4, a + 1)$]
]

#math.equation(block: true, alt: "the fraction 4 plus the fraction 4 over b minus 5 over the fraction 1 over b minus 5 plus the fraction b over 4")[$frac(4 + frac(4, b − 5), frac(1, b − 5) + frac(b, 4))$]

#strong[Simplify a Complex Rational Expression by Using the LCD]

In the following exercises, simplify each complex rational expression by using the LCD.

#math.equation(block: true, alt: "the fraction the fraction 1 over 3 plus the fraction 1 over 8 over the fraction 1 over 4 plus the fraction 1 over 12")[$frac(frac(1, 3) + frac(1, 8), frac(1, 4) + frac(1, 12))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 8")[$frac(11, 8)$]
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 4 plus the fraction 1 over 9 over the fraction 1 over 6 plus the fraction 1 over 12")[$frac(frac(1, 4) + frac(1, 9), frac(1, 6) + frac(1, 12))$]

#math.equation(block: true, alt: "the fraction the fraction 5 over 6 plus the fraction 2 over 9 over the fraction 7 over 18 minus the fraction 1 over 3")[$frac(frac(5, 6) + frac(2, 9), frac(7, 18) − frac(1, 3))$]

#solutionbox[
#math.equation(block: true, alt: "19")[$19$]
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 6 plus the fraction 4 over 15 over the fraction 3 over 5 minus the fraction 1 over 2")[$frac(frac(1, 6) + frac(4, 15), frac(3, 5) − frac(1, 2))$]

#math.equation(block: true, alt: "the fraction the fraction c over d plus the fraction 1 over d over the fraction 1 over d minus the fraction d over c")[$frac(frac(c, d) + frac(1, d), frac(1, d) − frac(d, c))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction c squared plus c over c minus d squared")[$frac(c^(2) + c, c − d^(2))$]
]

#math.equation(block: true, alt: "the fraction the fraction 1 over m plus the fraction m over n over the fraction n over m minus the fraction 1 over n")[$frac(frac(1, m) + frac(m, n), frac(n, m) − frac(1, n))$]

#math.equation(block: true, alt: "the fraction the fraction 1 over p plus the fraction 1 over q over the fraction 1 over p squared minus the fraction 1 over q squared")[$frac(frac(1, p) + frac(1, q), frac(1, p^(2)) − frac(1, q^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction p q over q minus p")[$frac(p q, q − p)$]
]

#math.equation(block: true, alt: "the fraction the fraction 2 over r plus the fraction 2 over t over the fraction 1 over r squared minus the fraction 1 over t squared")[$frac(frac(2, r) + frac(2, t), frac(1, r^(2)) − frac(1, t^(2)))$]

#math.equation(block: true, alt: "the fraction the fraction 2 over x plus 5 over the fraction 3 over x minus 5 plus the fraction 1 over x squared minus 25")[$frac(frac(2, x + 5), frac(3, x − 5) + frac(1, x^(2) − 25))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 x minus 10 over 3 x plus 16")[$frac(2 x − 10, 3 x + 16)$]
]

#math.equation(block: true, alt: "the fraction the fraction 5 over y minus 4 over the fraction 3 over y plus 4 plus the fraction 2 over y squared minus 16")[$frac(frac(5, y − 4), frac(3, y + 4) + frac(2, y^(2) − 16))$]

#math.equation(block: true, alt: "the fraction the fraction 5 over z squared minus 64 plus the fraction 3 over z plus 8 over the fraction 1 over z plus 8 plus the fraction 2 over z minus 8")[$frac(frac(5, z^(2) − 64) + frac(3, z + 8), frac(1, z + 8) + frac(2, z − 8))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 z minus 19 over 3 z plus 8")[$frac(3 z − 19, 3 z + 8)$]
]

#math.equation(block: true, alt: "the fraction the fraction 3 over s plus 6 plus the fraction 5 over s minus 6 over the fraction 1 over s squared minus 36 plus the fraction 4 over s plus 6")[$frac(frac(3, s + 6) + frac(5, s − 6), frac(1, s^(2) − 36) + frac(4, s + 6))$]

#math.equation(block: true, alt: "the fraction the fraction 4 over a squared minus 2 a minus 15 over the fraction 1 over a minus 5 plus the fraction 2 over a plus 3")[$frac(frac(4, a^(2) − 2 a − 15), frac(1, a − 5) + frac(2, a + 3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 3 a minus 7")[$frac(4, 3 a − 7)$]
]

#math.equation(block: true, alt: "the fraction the fraction 5 over b squared minus 6 b minus 27 over the fraction 3 over b minus 9 plus the fraction 1 over b plus 3")[$frac(frac(5, b^(2) − 6 b − 27), frac(3, b − 9) + frac(1, b + 3))$]

#math.equation(block: true, alt: "the fraction the fraction 5 over c plus 2 minus the fraction 3 over c plus 7 over the fraction 5 c over c squared plus 9 c plus 14")[$frac(frac(5, c + 2) − frac(3, c + 7), frac(5 c, c^(2) + 9 c + 14))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 c plus 29 over 5 c")[$frac(2 c + 29, 5 c)$]
]

#math.equation(block: true, alt: "the fraction the fraction 6 over d minus 4 minus the fraction 2 over d plus 7 over the fraction 2 d over d squared plus 3 d minus 28")[$frac(frac(6, d − 4) − frac(2, d + 7), frac(2 d, d^(2) + 3 d − 28))$]

#math.equation(block: true, alt: "the fraction 2 plus the fraction 1 over p minus 3 over the fraction 5 over p minus 3")[$frac(2 + frac(1, p − 3), frac(5, p − 3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 p minus 5 over 5")[$frac(2 p − 5, 5)$]
]

#math.equation(block: true, alt: "the fraction the fraction n over n minus 2 over 3 plus the fraction 5 over n minus 2")[$frac(frac(n, n − 2), 3 + frac(5, n − 2))$]

#math.equation(block: true, alt: "the fraction the fraction m over m plus 5 over 4 plus the fraction 1 over m minus 5")[$frac(frac(m, m + 5), 4 + frac(1, m − 5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction m open parenthesis m minus 5 close parenthesis over open parenthesis 4 m minus 19 close parenthesis open parenthesis m plus 5 close parenthesis")[$frac(m ( m − 5 ), ( 4 m − 19 ) ( m + 5 ))$]
]

#math.equation(block: true, alt: "the fraction 7 plus the fraction 2 over q minus 2 over the fraction 1 over q plus 2")[$frac(7 + frac(2, q − 2), frac(1, q + 2))$]

In the following exercises, simplify each complex rational expression using either method.

#math.equation(block: true, alt: "the fraction the fraction 3 over 4 minus the fraction 2 over 7 over the fraction 1 over 2 plus the fraction 5 over 14")[$frac(frac(3, 4) − frac(2, 7), frac(1, 2) + frac(5, 14))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 24")[$frac(13, 24)$]
]

#math.equation(block: true, alt: "the fraction the fraction v over w plus the fraction 1 over v over the fraction 1 over v minus the fraction v over w")[$frac(frac(v, w) + frac(1, v), frac(1, v) − frac(v, w))$]

#math.equation(block: true, alt: "the fraction the fraction 2 over a plus 4 over the fraction 1 over a squared minus 16")[$frac(frac(2, a + 4), frac(1, a^(2) − 16))$]

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis a minus 4 close parenthesis")[$2 ( a − 4 )$]
]

#math.equation(block: true, alt: "the fraction the fraction 3 over b squared minus 3 b minus 40 over the fraction 5 over b plus 5 minus the fraction 2 over b minus 8")[$frac(frac(3, b^(2) − 3 b − 40), frac(5, b + 5) − frac(2, b − 8))$]

#math.equation(block: true, alt: "the fraction the fraction 3 over m plus the fraction 3 over n over the fraction 1 over m squared minus the fraction 1 over n squared")[$frac(frac(3, m) + frac(3, n), frac(1, m^(2)) − frac(1, n^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 m n over n minus m")[$frac(3 m n, n − m)$]
]

#math.equation(block: true, alt: "the fraction the fraction 2 over r minus 9 over the fraction 1 over r plus 9 plus the fraction 3 over r squared minus 81")[$frac(frac(2, r − 9), frac(1, r + 9) + frac(3, r^(2) − 81))$]

#math.equation(block: true, alt: "the fraction x minus the fraction 3 x over x plus 2 over the fraction 3 over x plus 2 plus the fraction 3 over x minus 2")[$frac(x − frac(3 x, x + 2), frac(3, x + 2) + frac(3, x − 2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis x minus 1 close parenthesis open parenthesis x minus 2 close parenthesis over 6")[$frac(( x − 1 ) ( x − 2 ), 6)$]
]

#math.equation(block: true, alt: "the fraction the fraction y over y plus 3 over 2 plus the fraction 1 over y minus 3")[$frac(frac(y, y + 3), 2 + frac(1, y − 3))$]


==== Writing Exercises

In this section, you learned to simplify the complex fraction #math.equation(block: false, alt: "the fraction the fraction 3 over x plus 2 over the fraction x over x squared minus 4")[$frac(frac(3, x + 2), frac(x, x^(2) − 4))$] two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

#solutionbox[
Answers will vary.
]

Efraim wants to start simplifying the complex fraction #math.equation(block: false, alt: "the fraction the fraction 1 over a plus the fraction 1 over b over the fraction 1 over a minus the fraction 1 over b")[$frac(frac(1, a) + frac(1, b), frac(1, a) − frac(1, b))$] by cancelling the variables from the numerator and denominator, #math.equation(block: false, alt: "the fraction the fraction 1 over a plus the fraction 1 over b over the fraction 1 over a minus the fraction 1 over b .")[$frac(frac(1, cancel(a)) + frac(1, cancel(b)), frac(1, cancel(a)) − frac(1, cancel(b))) .$] Explain what is wrong with Efraim’s plan.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has four columns and three rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.], alt: "This table has four columns and three rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.", caption: none)

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?