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== 8.5#h(0.6em)Simplify Complex Rational Expressions

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

If you miss a problem, go back to the section listed and review the material.

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")[$frac(1, 54)$]
]
]

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we 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")[$frac(frac(3, 4), frac(5, 8)) #h(4em) frac(frac(x, 2), frac(x y, 6))$]In this section we will simplify #emph[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 or 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")[$frac(frac(4, y − 3), frac(8, y^(2) − 9)) #h(7em) frac(frac(1, x) + frac(1, y), frac(x, y) − frac(y, x)) #h(7em) frac(frac(2, x + 6), frac(4, x − 6) − frac(4, x^(2) − 36))$]Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.


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

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 rational expressions. We write it as if we were dividing two fractions.

#examplebox("Example 1")[][
Simplify: #math.equation(block: false, alt: "the fraction the fraction 4 over y minus 3 over the fraction 8 over y squared minus 9 .")[$frac(frac(4, y − 3), frac(8, y^(2) − 9)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the fraction 4 over y minus 3 over the fraction 8 over y squared minus 9")[$frac(frac(4, y − 3), frac(8, y^(2) − 9))$]]),
  [Rewrite the complex fraction as division.], [#math.equation(block: false, alt: "the fraction 4 over y minus 3 divided by the fraction 8 over y squared minus 9")[$frac(4, y − 3) ÷ frac(8, y^(2) − 9)$]],
  [Rewrite as the product of first times the #linebreak() reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 4 over y minus 3 times the fraction y squared minus 9 over 8")[$frac(4, y − 3) · frac(y^(2) − 9, 8)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 4 open parenthesis y squared minus 9 close parenthesis over 8 open parenthesis y minus 3 close parenthesis")[$frac(4 ( y^(2) − 9 ), 8 ( y − 3 ))$]],
  [Factor to look for common factors.], [#math.equation(block: false, alt: "the fraction 4 open parenthesis y minus 3 close parenthesis open parenthesis y plus 3 close parenthesis over 4 times 2 open parenthesis y minus 3 close parenthesis")[$frac(4 ( y − 3 ) ( y + 3 ), 4 · 2 ( y − 3 ))$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 4 open parenthesis y minus 3 close parenthesis open parenthesis y plus 3 close parenthesis over 4 times 2 open parenthesis y minus 3 close parenthesis")[$frac(cancel(4) cancel(( y − 3 )) ( y + 3 ), cancel(4) · 2 cancel(( y − 3 )))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction y plus 3 over 2")[$frac(y + 3, 2)$]],
))

Are there any value(s) of #math.equation(block: false, alt: "y")[$y$] that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original #strong[complex rational expression] had denominators of #math.equation(block: false, alt: "y minus 3")[$y − 3$] and #math.equation(block: false, alt: "y squared minus 9")[$y^(2) − 9$]. This expression would be undefined if #math.equation(block: false, alt: "y equals 3")[$y = 3$] or #math.equation(block: false, alt: "y equals −3")[$y = −3$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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: #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: #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 mathematical expression featuring a complex fraction. The numerator shows the sum of one-third and one-sixth, while the denominator displays the difference between one-half and one-third.], alt: "A mathematical expression featuring a complex fraction. The numerator shows the sum of one-third and one-sixth, while the denominator displays the difference between one-half and one-third.", caption: none)]),
  [Simplify the numerator and denominator.], [],
  [Find the LCD and add the fractions in the numerator. #linebreak() Find the LCD and add the fractions in the denominator.], [#figure(figph[A complex fraction showing addition and subtraction of fractions. Red numbers highlight the multiplication steps to find common denominators in both the numerator and denominator.], alt: "A complex fraction showing addition and subtraction of fractions. Red numbers highlight the multiplication steps to find common denominators in both the numerator and denominator.", caption: none)],
  [Simplify the numerator and denominator.], [#figure(figph[A complex fraction mathematical problem showing (2/6 + 1/6) divided by (3/6 - 2/6).], alt: "A complex fraction mathematical problem showing (2/6 + 1/6) divided by (3/6 - 2/6).", caption: none)],
  [Simplify the numerator and denominator, again.], [#figure(figph[A mathematical expression showing the division of two fractions, (3/6) by (1/6), which simplifies to 3. This illustrates how to divide fractions with a common denominator.], alt: "A mathematical expression showing the division of two fractions, (3/6) by (1/6), which simplifies to 3. This illustrates how to divide fractions with a common denominator.", caption: none)],
  [Rewrite the complex rational expression as a division problem.], [#figure(figph[A mathematical expression showing the division of two fractions: three-sixths divided by one-sixth.], alt: "A mathematical expression showing the division of two fractions: three-sixths divided by one-sixth.", caption: none)],
  [Multiply the first times by the reciprocal of the second.], [#figure(figph[A mathematical expression showing the multiplication of two fractions: 3/6 multiplied by 6/1.], alt: "A mathematical expression showing the multiplication of two fractions: 3/6 multiplied by 6/1.", caption: none)],
  [Simplify.], [#figure(figph[A close-up shot of the number '3' rendered in a simple, gray font against a clean white background. The number appears clearly centered, with a slight blur, indicating it might be a digital display or a simple graphic.], alt: "A close-up shot of the number '3' rendered in a simple, gray font against a clean white background. The number appears clearly centered, with a slight blur, indicating it might be a digital display or a simple graphic.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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: #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")[$frac(10, 23)$]
]
]

#examplebox("Example 3")[How to Simplify a Complex Rational Expression by Writing it as Division][
Simplify: #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[The above image has three columns. The image shows steps on how to divide complex rational expressions in three steps. Step one is to simplify the numerator and denominator. We will simplify the sum in the numerator and difference in the denominator for the example 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Find a common denominator and add the fractions in the numerator and find a common denominator and subtract the fractions in the numerator to get 1 times y divided by x times y plus 1 times x divided by y times x divided by x times x divided by y times x minus y times y divided by x times y. Then, we get y divided by x y plus x plus x y divided by x squared divided by x y minus y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator, y plus x divided by x y divided by x squared minus y squared divided by x y.], alt: "The above image has three columns. The image shows steps on how to divide complex rational expressions in three steps. Step one is to simplify the numerator and denominator. We will simplify the sum in the numerator and difference in the denominator for the example 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Find a common denominator and add the fractions in the numerator and find a common denominator and subtract the fractions in the numerator to get 1 times y divided by x times y plus 1 times x divided by y times x divided by x times x divided by y times x minus y times y divided by x times y. Then, we get y divided by x y plus x plus x y divided by x squared divided by x y minus y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator, y plus x divided by x y divided by x squared minus y squared divided by x y.", caption: none)

#figure(figph[Step two is to rewrite the complex rational expression as a division problem. We write the numerator divided by the denominator.], alt: "Step two is to rewrite the complex rational expression as a division problem. We write the numerator divided by the denominator.", caption: none)

#figure(figph[Step three is to divide the expressions. Multiply the first by the reciprocal of the second to get y plus x divided by x y times x y divided by x squared minus y squared. Factor any expressions if possible. We now have x y times y plus x divided by x y times x minus y times x plus y. Remove common factors. Cross out x, y and y plus x from the numerator. Cross out x, y and x plus y from the denominator. Simplify to get 1 divided by x minus y.], alt: "Step three is to divide the expressions. Multiply the first by the reciprocal of the second to get y plus x divided by x y times x y divided by x squared minus y squared. Factor any expressions if possible. We now have x y times y plus x divided by x y times x minus y times x plus y. Remove common factors. Cross out x, y and y plus x from the numerator. Cross out x, y and x plus y from the denominator. Simplify to get 1 divided by x minus y.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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: #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")[$frac(a b, b − a)$]
]
]

#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: #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 algebraic fraction involving variables 'n' and constants, with a numerator of (n - 4n/(n+5)) and a denominator of (1/(n+5) + 1/(n-5)).], alt: "A complex algebraic fraction involving variables 'n' and constants, with a numerator of (n - 4n/(n+5)) and a denominator of (1/(n+5) + 1/(n-5)).", caption: none)]),
  [Simplify the numerator and denominator.], [],
  [Find the LCD and add the fractions in the numerator. #linebreak() Find the LCD and add the fractions in the denominator.], [#figure(figph[An algebraic expression presented as a complex fraction. The numerator involves the subtraction of two rational terms, and the denominator involves the addition of two rational terms, all featuring the variable 'n'.], alt: "An algebraic expression presented as a complex fraction. The numerator involves the subtraction of two rational terms, and the denominator involves the addition of two rational terms, all featuring the variable 'n'.", caption: none)],
  [Simplify the numerators.], [#figure(figph[A complex algebraic fraction involving the variable 'n'. The numerator is the subtraction of (n^2+5n)/(n+5) and 4n/(n+5), all divided by the sum of (n-5)/((n+5)(n-5)) and (n+5)/((n-5)(n+5)).], alt: "A complex algebraic fraction involving the variable 'n'. The numerator is the subtraction of (n^2+5n)/(n+5) and 4n/(n+5), all divided by 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
add in the denominator. #linebreak()  #linebreak() Simplify.], [#figure(figph[A complex fraction with algebraic expressions, where the numerator is (n^2 + n)/(n + 5) and the denominator is 2n/((n + 5)(n - 5)).], alt: "A complex fraction with algebraic expressions, where the numerator is (n^2 + n)/(n + 5) and the denominator is 2n/((n + 5)(n - 5)).", caption: none)],
  [Rewrite as fraction division.], [#figure(figph[The image displays a mathematical expression: the fraction (n^2 + n) / (n + 5) divided by the fraction 2n / ((n + 5)(n - 5)).], alt: "The image displays a mathematical expression: the fraction (n^2 + n) / (n + 5) divided by the fraction 2n / ((n + 5)(n - 5)).", caption: none)],
  [Multiply the first times the reciprocal of the second.], [#figure(figph[An algebraic expression showing the product of two fractions. The first fraction is (n^2 + n) divided by (n + 5). The second fraction is ((n + 5)(n - 5)) divided by (2n).], alt: "An algebraic expression showing the product of two fractions. The first fraction is (n^2 + n) divided by (n + 5). The second fraction is ((n + 5)(n - 5)) divided by (2n).", caption: none)],
  [Factor any expressions if possible.], [#figure(figph[A mathematical expression showing a fraction with numerator n(n+1)(n+5)(n-5) and denominator (n+5)2n.], alt: "A mathematical expression showing a fraction with numerator n(n+1)(n+5)(n-5) and denominator (n+5)2n.", caption: none)],
  [Remove common factors.], [#figure(figph[A fractional algebraic expression with factors n, (n+1), (n+5), and (n-5) in the numerator, and (n+5) and 2n in the denominator, featuring cancellation marks for (n+5) and n.], alt: "A fractional algebraic expression with factors n, (n+1), (n+5), and (n-5) in the numerator, and (n+5) and 2n in the denominator, featuring cancellation marks for (n+5) and n.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction (n+1)(n-5) divided by 2. The numerator consists of the product of two binomials, (n+1) and (n-5), and the denominator is the number 2.], alt: "A mathematical expression showing the fraction (n+1)(n-5) divided by 2. The numerator consists of the product of two binomials, (n+1) and (n-5), and the denominator is the number 2.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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 b plus 2 b minus 5 over 3 b minus 5")[$frac(b b + 2 b − 5, 3 b − 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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 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: #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 mathematical expression: the sum of one-third and one-sixth, divided by the difference between one-half and one-third. It represents a multi-step fraction calculation.], alt: "A complex fraction mathematical expression: the sum of one-third and one-sixth, divided by the difference between one-half and one-third. It represents a multi-step fraction calculation.", caption: none)]),
  [The LCD of all the fractions in the whole expression is 6.], [],
  [Clear the fractions by multiplying the numerator and
denominator by that LCD.], [#figure(figph[Fraction calculation: (6 \* (1/3 + 1/6)) / (6 \* (1/2 - 1/3)), with the common multiplier '6' highlighted in red in both the numerator and denominator.], alt: "Fraction calculation: (6 * (1/3 + 1/6)) / (6 * (1/2 - 1/3)), with the common multiplier '6' highlighted in red in both the numerator and denominator.", caption: none)],
  [Distribute.], [#figure(figph[A complex fraction where the numerator is (6\*1/3 + 6\*1/6) and the denominator is (6\*1/2 - 6\*1/3), with the number 6 highlighted in red.], alt: "A complex fraction where the numerator is (6*1/3 + 6*1/6) and the denominator is (6*1/2 - 6*1/3), with the number 6 highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical fraction displays the expression (2 + 1) divided by (3 - 2) in black text on a white background.], alt: "A mathematical fraction displays the expression (2 + 1) divided by (3 - 2) in black text on a white background.", caption: none)],
  [], [#figure(figph[The image displays the fraction 3/1, with the number 3 positioned above a horizontal fraction bar and the number 1 directly below the bar, set against a plain white background.], alt: "The image displays the fraction 3/1, with the number 3 positioned above a horizontal fraction bar and the number 1 directly below the bar, set against a plain white background.", caption: none)],
  [], [#figure(figph[The numeral three, shown in a simple, grey font against a plain white background.], alt: "The numeral three, shown in a simple, grey font against a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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: #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")[$frac(10, 3)$]
]
]

#examplebox("Example 6")[How to Simplify a Complex Rational Expression by Using the LCD][
Simplify: #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[The above image has 3 columns. It shows the steps on how to simplify a complex rational expression using the LCD for 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Step one is to find the LCD of all fractions in the complex rational expression. The LCD of all the fractions is x y. Multiply the numerator and denominator by the LCD.], alt: "The above image has 3 columns. It shows the steps on how to simplify a complex rational expression using the LCD for 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Step one is to find the LCD of all fractions in the complex rational expression. The LCD of all the fractions is x y. Multiply the numerator and denominator by the LCD.", caption: none)

#figure(figph[Step two is to multiply both the numerator and denominator by x y to get x y times 1 divided by x plus 1 divided by y divided x y times x divided by y minus y divided by x.], alt: "Step two is to multiply both the numerator and denominator by x y to get x y times 1 divided by x plus 1 divided by y divided x y times x divided by y minus y divided by x.", caption: none)

#figure(figph[Step three is to simplify the expression. Distribute to get x y times 1 divided by x plus x y times 1 divided y divided by x y times x divided by y minus x y times y divided by x. Simplify to get y plus x divided by x squared minus y squared. Remove common factors. Cross out y plus x in the numerator. Cross out x plus y in the numerator. Simplify to get 1 divided by x minus y.], alt: "Step three is to simplify the expression. Distribute to get x y times 1 divided by x plus x y times 1 divided y divided by x y times x divided by y minus x y times y divided by x. Simplify to get y plus x divided by x squared minus y squared. Remove common factors. Cross out y plus x in the numerator. Cross out x plus y in the numerator. Simplify to get 1 divided by x minus y.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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: #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: #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 algebraic fraction with 2/(x+6) in the numerator and a difference of two fractions, 4/(x-6) - 4/(x^2-36), in the denominator.], alt: "A complex algebraic fraction with 2/(x+6) in the numerator and a difference of two fractions, 4/(x-6) - 4/(x^2-36), in the denominator.", caption: none)]),
  [Find the LCD of all fractions in the complex rational
expression. The LCD is #math.equation(block: false, alt: "open parenthesis x plus 6 close parenthesis open parenthesis x minus 6 close parenthesis")[$( x + 6 ) ( x − 6 )$].], [],
  [Multiply the numerator and denominator by the LCD.], [#figure(figph[A complex algebraic fraction demonstrating a step where both numerator and denominator are multiplied by the common factor (x+6)(x-6), highlighted in red, to simplify the expression.], alt: "A complex algebraic fraction demonstrating a step where both numerator and denominator are multiplied by the common factor (x+6)(x-6), highlighted in red, to simplify the expression.", caption: none)],
  [Simplify the expression.], [],
  [Distribute in the denominator.], [#figure(figph[A step in simplifying a rational expression, illustrating the multiplication of the numerator and denominator by the common factor (x+6)(x-6) to remove inner fractions.], alt: "A step in simplifying a rational expression, illustrating the multiplication of the numerator and denominator by the common factor (x+6)(x-6) to remove inner fractions.", caption: none)],
  [Simplify.], [#figure(figph[Algebraic expression being simplified, demonstrating cancellation of common factors (x+6) and (x-6) in numerator and denominator to reduce the complex fraction.], alt: "Algebraic expression being simplified, demonstrating cancellation of common factors (x+6) and (x-6) in numerator and denominator to reduce the complex fraction.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical fraction is shown. The numerator is 2(x - 6) and the denominator is 4(x + 6) - 4.], alt: "A mathematical fraction is shown. The numerator is 2(x - 6) and the denominator is 4(x + 6) - 4.", caption: none)],
  [To simplify the denominator, distribute
and combine like terms.], [#figure(figph[A mathematical expression shown as a fraction with 2(x-6) in the numerator and 4x+20 in the denominator, set against a plain white background.], alt: "A mathematical expression shown as a fraction with 2(x-6) in the numerator and 4x+20 in the denominator, set against a plain white background.", caption: none)],
  [Remove common factors.], [#figure(figph[A mathematical expression showing a fraction with 2(x-6) in the numerator and 2(2x+10) in the denominator.], alt: "A mathematical expression showing a fraction with 2(x-6) in the numerator and 2(2x+10) in the denominator.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction (x-6) over (2x+10).], alt: "A mathematical expression showing the fraction (x-6) over (2x+10).", caption: none)],
  [Notice that there are no more factors
common to the numerator and denominator.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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 x minus 6 over 5 x plus 7")[$frac(3 x − 6, 5 x + 7)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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")[$frac(x + 21, 6 x − 43)$]
]
]

#examplebox("Example 8")[][
Simplify: #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. The numerator is 4, and the denominator is another fraction with m^2 - 7m + 12 as its numerator and (3/(m-3) - 2/(m-4)) as its denominator.], alt: "A complex algebraic fraction. The numerator is 4, and the denominator is another fraction with m^2 - 7m + 12 as its numerator and (3/(m-3) - 2/(m-4)) as its denominator.", caption: none)]),
  [Find the LCD of all fractions in the
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 denominator by the LCD.], [#figure(figph[A step in simplifying a complex algebraic fraction by multiplying by the common denominator (m-3)(m-4) to clear fractions in both the numerator and denominator.], alt: "A step in simplifying a complex algebraic fraction by multiplying by the common denominator (m-3)(m-4) to clear fractions in both the numerator and denominator.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression is shown, demonstrating the cancellation of common factors (m-3) and (m-4) with red strikethroughs in both the numerator and the denominator to simplify the rational expression.], alt: "A mathematical expression is shown, demonstrating the cancellation of common factors (m-3) and (m-4) with red strikethroughs in both the numerator and the denominator to simplify the rational expression.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression featuring a fraction with '4' as the numerator and '3(m-4) - 2(m-3)' as the denominator, suitable for algebraic simplification.], alt: "A mathematical expression featuring a fraction with '4' as the numerator and '3(m-4) - 2(m-3)' as the denominator, suitable for algebraic simplification.", caption: none)],
  [Distribute.], [#figure(figph[The image shows a mathematical fraction where the numerator is 4 and the denominator is 3m - 12 - 2m + 6.], alt: "The image shows a mathematical fraction where the numerator is 4 and the denominator is 3m - 12 - 2m + 6.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical expression showing the fraction 4 over (m minus 6).], alt: "A mathematical expression showing the fraction 4 over (m minus 6).", caption: none)],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 y plus 34 over 3 y")[$frac(6 y + 34, 3 y)$]
]
]

#examplebox("Example 9")[][
Simplify: #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 the numerator as y divided by (y+1) and the denominator as 1 plus 1 divided by (y-1).], alt: "A complex fraction with the numerator as y divided by (y+1) and the denominator as 1 plus 1 divided by (y-1).", 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 fraction where both numerator and denominator are multiplied by the red terms (y+1)(y-1). The expression involves a fraction y/(y+1) in the numerator and (1 + 1/(y-1)) in the denominator.], alt: "An algebraic fraction where both numerator and denominator are multiplied by the red terms (y+1)(y-1). The expression involves a fraction y/(y+1) in the numerator and (1 + 1/(y-1)) in the denominator.", caption: none)],
  [Distribute in the denominator and simplify.], [#figure(figph[A mathematical expression featuring algebraic terms (y+1) and (y-1) in both the numerator and denominator, with red strike-throughs indicating the cancellation of common factors during simplification.], alt: "A mathematical expression featuring algebraic terms (y+1) and (y-1) in both the numerator and denominator, with red strike-throughs indicating the cancellation of common factors during simplification.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing a fraction. The numerator is (y-1)y. The denominator is (y+1)(y-1) + (y+1).], alt: "A mathematical expression showing a fraction. The numerator is (y-1)y. The denominator is (y+1)(y-1) + (y+1).", caption: none)],
  [Simplify the denominator, and leave the numerator
factored.], [#figure(figph[A mathematical fraction with y(y-1) in the numerator and y^2 - 1 + y + 1 in the denominator, set against a plain white background.], alt: "A mathematical fraction with y(y-1) in the numerator and y^2 - 1 + y + 1 in the denominator, set against a plain white background.", caption: none)],
  [], [#figure(figph[A mathematical expression showing a fraction with y(y-1) in the numerator and y^2 + y in the denominator, written in black text on a white background.], alt: "A mathematical expression showing a fraction with y(y-1) in the numerator and y^2 + y in the denominator, written in black text on a white background.", caption: none)],
  [Factor the denominator, and remove factors common
with the numerator.], [#figure(figph[A mathematical expression showing the fraction x(y-1) over x(y+1), where the 'x' terms are crossed out, indicating simplification to (y-1) over (y+1).], alt: "A mathematical expression showing the fraction x(y-1) over x(y+1), where the 'x' terms are crossed out, indicating simplification to (y-1) over (y+1).", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the fraction (y - 1) / (y + 1).], alt: "A mathematical expression showing the fraction (y - 1) / (y + 1).", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #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: #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")[$frac(x ( x + 1 ), 3 ( x − 1 ))$]
]
]


=== Key Concepts

- #strong[To Simplify a 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[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.

#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 minus 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 minus 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.

#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 open parenthesis 2 p minus 5 close parenthesis 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 4 m squared plus m minus 95")[$frac(m ( m − 5 ), 4 m^(2) + m − 95)$]
]

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

#strong[Simplify]

In the following exercises, use 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")[$frac(frac(y, y + 3), 2 + frac(1, y − 3))$]


==== Everyday Math

#strong[Electronics] The resistance of a circuit formed by connecting two resistors in parallel is #math.equation(block: false, alt: "the fraction 1 over the fraction 1 over R sub 1 plus the fraction 1 over R sub 2")[$frac(1, frac(1, R_(1)) + frac(1, R_(2)))$].

+ ⓐ Simplify the complex fraction #math.equation(block: false, alt: "the fraction 1 over the fraction 1 over R sub 1 plus the fraction 1 over R sub 2")[$frac(1, frac(1, R_(1)) + frac(1, R_(2)))$].
+ ⓑ Find the resistance of the circuit when #math.equation(block: false, alt: "R sub 1 equals 8")[$R_(1) = 8$] and #math.equation(block: false, alt: "R sub 2 equals 12")[$R_(2) = 12$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction R sub 1 R sub 2 over R sub 2 plus R sub 1")[$frac(R_(1) R_(2), R_(2) + R_(1))$] ⓑ #math.equation(block: false, alt: "the fraction 24 over 5")[$frac(24, 5)$]
]

#strong[Ironing] Lenore can do the ironing for her family’s business in #math.equation(block: false, alt: "h")[$h$] hours. Her daughter would take #math.equation(block: false, alt: "h plus 2")[$h + 2$] hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is #math.equation(block: false, alt: "the fraction 1 over the fraction 1 over h plus the fraction 1 over h plus 2")[$frac(1, frac(1, h) + frac(1, h + 2))$].

+ ⓐ Simplify the complex fraction #math.equation(block: false, alt: "the fraction 1 over the fraction 1 over h plus the fraction 1 over h plus 2")[$frac(1, frac(1, h) + frac(1, h + 2))$].
+ ⓑ Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if #math.equation(block: false, alt: "h equals 4")[$h = 4$].


==== 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

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. 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[The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.], alt: "The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.", caption: none)

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