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== 10.9#h(0.6em)Working with Algebraic Fractions


    A quotient of two polynomials is called a #strong[rational expression] or an #strong[algebraic fraction]. Operations on algebraic fractions follow the same rules as operations on common fractions.


    
      
=== Reducing Fractions


      When we reduce an ordinary fraction such as #math.equation(block: false, alt: "the fraction 24 over 36")[$display(frac(24, 36))$], we are using the fundamental principle of fractions.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fundamental Principle of Fractions]


        If we multiply or divide the numerator and denominator of a fraction by the same (nonzero) number, the new fraction is equivalent to the old one. In symbols,


        #math.equation(block: true, alt: "the fraction a c over b c equals the fraction a over b , open parenthesis b , c not equal to 0 close parenthesis")[$frac(a c, b c) = frac(a, b) , #hide($0000$) ( b , " " c ≠ 0 )$]
]


      Thus, for example,


      #math.equation(block: true, alt: "the fraction 24 over 36 equals the fraction 2 times 12 over 3 times 12 equals the fraction 2 over 3")[$frac(24, 36) = frac(2 ⋅ 12, 3 ⋅ 12) = frac(2, 3)$]
      We use the same procedure to reduce algebraic fractions: We look for common factors in the numerator and denominator and then apply the fundamental principle.


      #examplebox("Example 1")[][
Reduce each algebraic fraction.


          + #math.equation(block: false, alt: "the fraction 8 x cubed y over 6 x squared y cubed")[$display(frac(8 x^(3) y, 6 x^(2) y^(3)))$]
+ #math.equation(block: false, alt: "the fraction 6 x minus 3 over 3")[$display(frac(6 x − 3, 3))$]


        
        #solutionbox[
Factor out any common factors from the numerator and denominator. Then divide numerator and denominator by the common factors.


          + #math.equation(block: false, alt: "the fraction 8 x cubed y over 6 x squared y cubed equals the fraction 4 x times 2 x squared y over 3 y squared times 2 x squared y equals the fraction 4 x over 3 y squared")[$display(frac(8 x^(3) y, 6 x^(2) y^(3))) = display(frac(4 x ⋅ 2 x^(2) y, 3 y^(2) ⋅ 2 x^(2) y)) = display(frac(4 x, 3 y^(2)))$]
+ #math.equation(block: false, alt: "the fraction 6 x minus 3 over 3 equals the fraction 3 open parenthesis 2 x plus 1 close parenthesis over 3 equals 2 x plus 1")[$display(frac(6 x − 3, 3)) = display(frac(cancel(3) ( 2 x + 1 ), cancel(3))) = 2 x + 1$]
]
]


      If the numerator or denominator of the fraction contains more than one term, it is especially important to #emph[factor] before attempting to apply the fundamental principle. We can divide out common #emph[factors] from the numerator and denominator of a fraction, but the fundamental principle does #emph[not] apply to common #emph[terms].


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
We can reduce


        #math.equation(block: true, alt: "the fraction 2 x y over 3 y equals the fraction 2 x over 3")[$frac(2 x y, 3 y) = frac(2 x, 3)$]
        because #math.equation(block: false, alt: "y")[$y$] is a common factor in the numerator and denominator. However,


        #math.equation(block: true, alt: "the fraction 2 x plus y over 3 plus y not equal to the fraction 2 x over 3")[$frac(2 x + y, 3 + y) ≠ frac(2 x, 3)$]
        because #math.equation(block: false, alt: "y")[$y$] is a common term but is #emph[not a common factor] of the numerator and denominator. Furthermore,


        #math.equation(block: true, alt: "the fraction 5 x plus 3 over 5 y not equal to the fraction x plus 3 over y")[$frac(5 x + 3, 5 y) ≠ frac(x + 3, y)$]
        because #math.equation(block: false, alt: "5")[$5$] is not a factor of the #emph[entire] numerator.
]


      #examplebox("Example 2")[][
Reduce each fraction.


          + #math.equation(block: false, alt: "the fraction 4 x plus 2 over 4")[$display(frac(4 x + 2, 4))$]
+ #math.equation(block: false, alt: "the fraction 9 x squared plus 3 over 6 x plus 3")[$display(frac(9 x^(2) + 3, 6 x + 3))$]


        
        #solutionbox[
Factor the numerator and denominator. Then divide numerator and denominator by the common factors.


          + #math.equation(block: false, alt: "the fraction 4 x plus 2 over 4 equals the fraction 2 open parenthesis 2 x plus 1 close parenthesis over 2 open parenthesis 2 close parenthesis the fraction 2 x plus 1 over 2")[$display(frac(4 x + 2, 4)) = display(frac(cancel(2) ( 2 x + 1 ), cancel(2) ( 2 ))) display(frac(2 x + 1, 2))$]
+ #math.equation(block: false, alt: "the fraction 9 x squared plus 3 over 6 x plus 3 equals the fraction 3 open parenthesis 3 x squared plus 1 close parenthesis over 3 open parenthesis 2 x plus 1 close parenthesis equals the fraction 3 x squared plus 1 over 2 x plus 1")[$display(frac(9 x^(2) + 3, 6 x + 3)) = display(frac(cancel(3) ( 3 x^(2) + 1 ), cancel(3) ( 2 x + 1 ))) = display(frac(3 x^(2) + 1, 2 x + 1))$]
]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Note that in Examplea above,


        #math.equation(block: true, alt: "the fraction 4 x plus 2 over 4 not equal to x plus 2")[$frac(4 x + 2, 4) ≠ x + 2$]
        and in Exampleb,


        #math.equation(block: true, alt: "the fraction 9 x squared plus 3 over 6 x plus 3 not equal to the fraction 9 x squared over 6 x")[$frac(9 x^(2) + 3, 6 x + 3) ≠ frac(9 x^(2), 6 x)$]
]


      We summarize the procedure for reducing algebraic fractions as follows.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Reduce an Algebraic Fraction:]


        + Factor the numerator and denominator.
+ Divide the numerator and denominator by any common factors.
]


      #examplebox("Example 3")[][
Reduce each fraction.


          + #math.equation(block: false, alt: "the fraction x squared minus 7 x plus 6 over 36 minus x squared")[$display(frac(x^(2) − 7 x + 6, 36 − x^(2)))$]
+ #math.equation(block: false, alt: "the fraction 27 x cubed minus 1 over 9 x squared minus 1")[$display(frac(27 x^(3) − 1, 9 x^(2) − 1))$]


        
        #solutionbox[
+ Factor numerator and denominator to obtain
                #math.equation(block: true, alt: "the fraction open parenthesis x minus 6 close parenthesis open parenthesis x minus 1 close parenthesis over open parenthesis 6 minus x close parenthesis open parenthesis 6 plus x close parenthesis")[$frac(( x − 6 ) ( x − 1 ), ( 6 − x ) ( 6 + x ))$]
            The factor #math.equation(block: false, alt: "x minus 6")[$x − 6$] in the numerator is the opposite of the factor #math.equation(block: false, alt: "6 minus x")[$6 − x$] in the denominator. That is, #math.equation(block: false, alt: "x minus 6 equals minus 1 open parenthesis 6 minus x close parenthesis")[$x − 6 = − 1 ( 6 − x )$]. Thus,
                #math.equation(block: true, alt: "the fraction minus 1 open parenthesis 6 minus x close parenthesis open parenthesis x minus 1 close parenthesis over open parenthesis 6 minus x close parenthesis open parenthesis 6 plus x close parenthesis equals the fraction minus 1 open parenthesis x minus 1 close parenthesis over 6 plus x equals the fraction 1 minus x over 6 plus x")[$frac(− 1 cancel(( 6 − x )) ( x − 1 ), cancel(( 6 − x )) ( 6 + x )) = frac(− 1 ( x − 1 ), 6 + x) = frac(1 − x, 6 + x)$]
+ The numerator of the fraction is a difference of two cubes, and the denominator is a difference of two squares. Factor each to obtain
                #math.equation(block: true, alt: "the fraction open parenthesis 3 x minus 1 close parenthesis open parenthesis 9 x squared plus 3 x plus 1 close parenthesis over open parenthesis 3 x minus 1 close parenthesis open parenthesis 3 x plus 1 close parenthesis equals the fraction 9 x squared plus 3 x plus 1 over 3 x plus 1")[$frac(cancel(( 3 x − 1 )) ( 9 x^(2) + 3 x + 1 ), cancel(( 3 x − 1 )) ( 3 x + 1 )) = frac(9 x^(2) + 3 x + 1, 3 x + 1)$]
]
]


    
    
      
=== Products of Fractions


      To multiply two or more common fractions together, we multiply their numerators together and multiply their denominators together. The same is true for a product of algebraic fractions. For example, xy


      #math.equation(block: true, alt: "the fraction 6 x squared over y times the fraction x y over 2, equals the fraction 6 x squared times over y times 2 equals the fraction 6 x cubed y over 2 y, Reduce.; equals the fraction 3 x cubed open parenthesis 2 y close parenthesis over 2 y equals 3 x cubed")[$frac(6 x^(2), y) ⋅ frac(x y, 2) & = frac(6 x^(2) ⋅, y ⋅ 2) = frac(6 x^(3) y, 2 y) &  & "Reduce." \  & = frac(3 x^(3) ( 2 y ), 2 y) = 3 x^(3)$]
      We can simplify the process by first factoring each numerator and denominator and dividing out any common factors.


      #math.equation(block: true, alt: "the fraction 6 x squared over y times the fraction 2 times 3 x squared over y times the fraction x y over 2 equals 3 x cubed")[$frac(6 x^(2), y) ⋅ frac(cancel(2) ⋅ 3 x^(2), cancel(y)) ⋅ frac(x cancel(y), cancel(2)) = 3 x^(3)$]
      In general, we have the following procedure for finding the product of algebraic fractions.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Multiply Algebraic Fractions:]


        + Factor each numerator and denominator.
+ Divide out any factors that appear in both a numerator and a denominator.
+ Multiply together the numerators; multiply together the denominators.
]


      #examplebox("Example 4")[][
Find each product.


          + #math.equation(block: false, alt: "the fraction 5 over x squared minus 1 times the fraction x plus 2 over x")[$display(frac(5, x^(2) − 1)) ⋅ display(frac(x + 2, x))$]
+ #math.equation(block: false, alt: "the fraction 4 y squared minus 1 over 4 minus y squared times the fraction y squared minus 2 y over 4 y plus 2")[$display(frac(4 y^(2) − 1, 4 − y^(2))) ⋅ display(frac(y^(2) − 2 y, 4 y + 2))$]


        
        #solutionbox[
+ The denominator of the first fraction factors into #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis open parenthesis x minus 1 close parenthesis")[$( x + 1 ) ( x − 1 )$]. There are no common factors to divide out, so we multiply the numerators together and multiply the denominators together.
                #math.equation(block: true, alt: "the fraction 5 over x squared minus 1 times the fraction x plus 2 over x equals the fraction 5 open parenthesis x plus 2 close parenthesis over x open parenthesis x squared minus 1 close parenthesis equals the fraction 5 x plus 10 over x cubed minus x")[$frac(5, x^(2) − 1) ⋅ frac(x + 2, x) = frac(5 ( x + 2 ), x ( x^(2) − 1 )) = frac(5 x + 10, x^(3) − x)$]
+ Factor each numerator and each denominator. Look for common factors.
                #math.equation(block: true, alt: "the fraction 4 y squared minus 1 over 4 minus y squared times the fraction y squared minus 2 y over 4 y plus 2, equals the fraction open parenthesis 2 y minus 1 close parenthesis open parenthesis 2 y plus 1 close parenthesis over open parenthesis 2 minus y close parenthesis open parenthesis 2 plus y close parenthesis times the fraction y open parenthesis y minus 2 close parenthesis to the power minus 1 over 2 open parenthesis 2 y plus 1 close parenthesis, Divide out common factors.; equals the fraction minus y open parenthesis 2 y minus 1 close parenthesis over 2 open parenthesis y plus 2 close parenthesis, Note: y minus 2 equals minus open parenthesis 2 minus y close parenthesis")[$frac(4 y^(2) − 1, 4 − y^(2)) ⋅ frac(y^(2) − 2 y, 4 y + 2) & = frac(( 2 y − 1 ) ( cancel(2 y + 1) ), cancel(( 2 − y )) ( 2 + y )) ⋅ frac(y attach(cancel(( y − 2 )), t: − 1), 2 ( cancel(2 y + 1) )) &  & "Divide out common factors." \  & = frac(− y ( 2 y − 1 ), 2 ( y + 2 )) &  & "Note: " " " bold(italic(y)) − 2 = − ( 2 − bold(italic(y)) )$]
]
]


    
    
      
=== Quotients of Fractions


      To divide two algebraic fractions we multiply the first fraction by the reciprocal of the second fraction. For example,


      #math.equation(block: true, alt: "the fraction 2 x cubed over 3 y divided by the fraction 4 x over 5 y squared, equals the fraction 2 x cubed over 3 y times the fraction 5 y squared over 4 x; equals the fraction 2 x times x squared over 3 y times the fraction 5 y times y over 2 times 2 x equals the fraction 5 x squared y over 6")[$frac(2 x^(3), 3 y) ÷ frac(4 x, 5 y^(2)) & = frac(2 x^(3), 3 y) ⋅ frac(5 y^(2), 4 x) \  & = frac(cancel(2 x) ⋅ x^(2), 3 cancel(y)) ⋅ frac(5 y ⋅ cancel(y), 2 ⋅ cancel(2 x)) = frac(5 x^(2) y, 6)$]
      If the fractions involve polynomials of more than one term, we may need to factor each numerator and denominator in order to recognize any common factors. This suggests the following procedure for dividing algebraic fractions.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Divide Algebraic Fractions:]


        + Multiply the first fraction by the reciprocal of the second fraction.
+ Factor each numerator and denominator.
+ Divide out any factors that appear in both a numerator and a denominator.
+ Multiply together the numerators; multiply together the denominators.
]


      #examplebox("Example 5")[][
Find each quotient.


          + #math.equation(block: false, alt: "the fraction x squared minus 1 over x plus 3 divided by the fraction x squared minus x minus 2 over x squared plus 5 x plus 6")[$display(frac(x^(2) − 1, x + 3)) ÷ display(frac(x^(2) − x − 2, x^(2) + 5 x + 6))$]
+ #math.equation(block: false, alt: "the fraction 6 a b over 2 a plus b divided by open parenthesis 4 a squared b close parenthesis")[$display(frac(6 a b, 2 a + b)) ÷ ( 4 a^(2) b )$]


        
        #solutionbox[
+ Multiply the first fraction by the reciprocal of the second fraction.
            #math.equation(block: true, alt: "the fraction x squared minus 1 over x plus 3 divided by the fraction x squared minus x minus 2 over x squared plus 5 x plus 6, equals the fraction x squared minus 1 over x plus 3 times the fraction x squared plus 5 x plus 6 over x squared minus x minus 2, Factor.; equals the fraction open parenthesis x minus 1 close parenthesis open parenthesis x plus 1 close parenthesis over x plus 3 times the fraction open parenthesis x plus 3 close parenthesis open parenthesis x plus 2 close parenthesis over open parenthesis x plus 1 close parenthesis open parenthesis x minus 2 close parenthesis; equals the fraction open parenthesis x minus 1 close parenthesis open parenthesis x plus 2 close parenthesis over x minus 2")[$frac(x^(2) − 1, x + 3) ÷ frac(x^(2) − x − 2, x^(2) + 5 x + 6) & = frac(x^(2) − 1, x + 3) ⋅ frac(x^(2) + 5 x + 6, x^(2) − x − 2) &  & "Factor." \  & = frac(( x − 1 ) ( cancel(x + 1) ), cancel(x + 3)) ⋅ frac(( cancel(x + 3) ) ( x + 2 ), ( cancel(x + 1) ) ( x − 2 )) \  & = frac(( x − 1 ) ( x + 2 ), x − 2)$]
+ Multiply the first fraction by the reciprocal of the second fraction.
            #math.equation(block: true, alt: "the fraction 6 a b over 2 a plus b divided by open parenthesis 4 a squared b close parenthesis, equals the fraction 6 cubed a b over 2 a plus b times the fraction 1 over 4 squared a times a b, Divide out common factors.; equals the fraction 3 over 2 a open parenthesis 2 a plus b close parenthesis equals the fraction 3 over 4 a squared plus 2 a b")[$frac(6 a b, 2 a + b) ÷ ( 4 a^(2) b ) & = frac(cancel(6)^(3) cancel(a b), 2 a + b) ⋅ frac(1, cancel(4)^(2) a ⋅ cancel(a b)) &  & "Divide out common factors." \  & = frac(3, 2 a ( 2 a + b )) = frac(3, 4 a^(2) + 2 a b)$]
]
]


    
    
      
=== Sums and Differences of Like Fractions


      Algebraic fractions with the same denominator are called #strong[like fractions]. To add or subtract like fractions, we combine their numerators and keep the same denominator for the sum or difference. This method is an application of the distributive law.


      #examplebox("Example 6")[][
Find each sum or difference.


          + #math.equation(block: false, alt: "the fraction 2 x over 9 z squared plus the fraction 5 x over 9 z squared")[$display(frac(2 x, 9 z^(2))) + display(frac(5 x, 9 z^(2)))$]
+ #math.equation(block: false, alt: "the fraction 2 x minus 1 over x plus 3 minus the fraction 5 x minus 3 over x plus 3")[$display(frac(2 x − 1, x + 3)) − display(frac(5 x − 3, x + 3))$]


        
        #solutionbox[
+ Because these are like fractions, we add their numerators and keep the same denominator.
                #math.equation(block: true, alt: "the fraction 2 x over 9 z squared plus the fraction 5 x over 9 z squared equals the fraction 2 x plus 5 x over 9 z squared equals the fraction 7 x over 9 z squared")[$frac(2 x, 9 z^(2)) + frac(5 x, 9 z^(2)) = frac(2 x + 5 x, 9 z^(2)) = frac(7 x, 9 z^(2))$]
+ Be careful to subtract the #emph[entire] numerator of the second fraction: Use parentheses to show that the subtraction applies to both terms of #math.equation(block: false, alt: "5 x minus 3")[$5 x − 3$].
                #math.equation(block: true, alt: "the fraction 2 x minus 1 over x plus 3 minus the fraction 5 x minus 3 over x plus 3, equals the fraction 2 x minus 1 minus open parenthesis 5 x minus 3 close parenthesis over x plus 3; equals the fraction 2 x minus 1 minus 5 x plus 3 over x plus 3 equals the fraction minus 3 x plus 2 over x plus 3")[$frac(2 x − 1, x + 3) − frac(5 x − 3, x + 3) & = frac(2 x − 1 − ( 5 bold(italic(x)) − 3 ), x + 3) \  & = frac(2 x − 1 − 5 bold(italic(x)) + 3, x + 3) = frac(− 3 x + 2, x + 3)$]
]
]


    
    
      
=== Lowest Common Denominator


      To add or subtract fractions with different denominators, we must first find a #strong[common denominator].


      For arithmetic fractions, we use the smallest natural number that is exactly divisible by each of the given denominators. For example, to add the fractions #math.equation(block: false, alt: "the fraction 1 over 6")[$display(frac(1, 6))$] and #math.equation(block: false, alt: "the fraction 3 over 8")[$display(frac(3, 8))$], we use #math.equation(block: false, alt: "24")[$24$] as the common denominator because #math.equation(block: false, alt: "24")[$24$] is the smallest natural number that both #math.equation(block: false, alt: "6")[$6$] and #math.equation(block: false, alt: "8")[$8$] divide into evenly.


      We define the #strong[lowest common denominator] (#strong[LCD]) of two or more algebraic fractions as the polynomial of least degree that is exactly divisible by each of the given denominators.


      #examplebox("Example 7")[][
Find the LCD for the fractions #math.equation(block: false, alt: "the fraction 3 x over x plus 2")[$display(frac(3 x, x + 2))$] and #math.equation(block: false, alt: "the fraction 2 x over x minus 3")[$display(frac(2 x, x − 3))$]


        
        #solutionbox[
The LCD is a polynomial that has as factors both #math.equation(block: false, alt: "x plus 2")[$x + 2$] and #math.equation(block: false, alt: "x minus 3")[$x − 3$]. The simplest such polynomial is #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x minus 3 close parenthesis")[$( x + 2 ) ( x − 3 )$], or #math.equation(block: false, alt: "x squared minus x minus 6")[$x^(2) − x − 6$]. For our purposes, it will be more convenient to leave the LCD in factored form, so the LCD is #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x minus 3 close parenthesis")[$( x + 2 ) ( x − 3 )$].
]
]


      The LCD in Example was easy to find because each original denominator consisted of a single factor; that is, neither denominator could be factored. In that case, the LCD is just the product of the original denominators.


      We can always find a common denominator by multiplying together all the denominators in the given fractions, but this may not give us the #emph[simplest] or #emph[lowest] common denominator. Using anything other than the simplest possible common denominator will complicate our work needlessly.


      If any of the denominators in the given fractions can be factored, we factor them before looking for the LCD.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Find the LCD of Algebraic Fractions:]


        + Factor each denominator completely.
+ Include each different factor in the LCD as many times as it occurs in any #emph[one] of the given denominators.
]


      #examplebox("Example 8")[][
Find the LCD for the fractions #math.equation(block: false, alt: "the fraction 2 x over x squared minus 1")[$display(frac(2 x, x^(2) − 1))$] and #math.equation(block: false, alt: "the fraction x plus 3 over x squared plus x")[$display(frac(x + 3, x^(2) + x))$].


        
        #solutionbox[
Factor the denominators of each of the given fractions.


          #math.equation(block: true, alt: "x squared minus 1 equals open parenthesis x minus 1 close parenthesis open parenthesis x plus 1 close parenthesis and x squared plus x equals x open parenthesis x plus 1 close parenthesis")[$x^(2) − 1 = ( x − 1 ) ( x + 1 ) " " " " " " " and " " " " " " " x^(2) + x = x ( x + 1 )$]
          The factor #math.equation(block: false, alt: "open parenthesis x minus 1 close parenthesis")[$( x − 1 )$] occurs once in the first denominator, the factor #math.equation(block: false, alt: "x")[$x$] occurs once in the second denominator, and the factor #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis")[$( x + 1 )$] occurs once in each denominator. Therefore, we include in our LCD one copy of each of these factors. The LCD is #math.equation(block: false, alt: "x open parenthesis x plus 1 close parenthesis open parenthesis x minus 1 close parenthesis")[$x ( x + 1 ) ( x − 1 )$].
]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In Example, we do not include two factors of #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis")[$( x + 1 )$] in the LCD. We need only one factor of #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis")[$( x + 1 )$] because #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis")[$( x + 1 )$] occurs only once in either denominator. You should check that each original denominator divides evenly into our LCD, #math.equation(block: false, alt: "x open parenthesis x plus 1 close parenthesis open parenthesis x minus 1 close parenthesis")[$x ( x + 1 ) ( x − 1 )$].
]


    
    
      
=== Building Fractions


      After finding the LCD, we #strong[build] each fraction to an equivalent one with the LCD as its denominator. The new fractions will be like fractions, and we can combine them as explained above.


      Building a fraction is the opposite of reducing a fraction, in the sense that we multiply, rather than divide, the numerator and denominator by an appropriate factor. To find the #strong[building factor], we compare the factors of the original denominator with those of the desired common denominator.


      #examplebox("Example 9")[][
Build each of the fractions #math.equation(block: false, alt: "the fraction 3 x over x plus 2")[$display(frac(3 x, x + 2))$] and #math.equation(block: false, alt: "the fraction 2 x over x minus 3")[$display(frac(2 x, x − 3))$] to equivalent fractions with the LCD #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x minus 3 close parenthesis")[$( x + 2 ) ( x − 3 )$] as denominator.


        
        #solutionbox[
Compare the denominator of the given fraction to the LCD. We see that the fraction #math.equation(block: false, alt: "the fraction 3 x over x plus 2")[$display(frac(3 x, x + 2))$] needs a factor of #math.equation(block: false, alt: "open parenthesis x minus 3 close parenthesis")[$( x − 3 )$] in its denominator, so #math.equation(block: false, alt: "open parenthesis x minus 3 close parenthesis")[$( x − 3 )$] is the building factor for the first fraction. We multiply the numerator and denominator of the first fraction by #math.equation(block: false, alt: "open parenthesis x minus 3 close parenthesis")[$( x − 3 )$] to obtain an equivalent fraction:


          #math.equation(block: true, alt: "the fraction 3 x over x plus 2 equals the fraction 3 x open parenthesis x minus 3 close parenthesis over open parenthesis x plus 2 close parenthesis open parenthesis x minus 3 close parenthesis equals the fraction 3 x squared minus 9 x over x squared minus x minus 6")[$frac(3 x, x + 2) = frac(3 x ( bold(italic(x)) − 3 ), ( x + 2 ) ( bold(italic(x)) − 3 )) = frac(3 x^(2) − 9 x, x^(2) − x − 6)$]
          The fraction #math.equation(block: false, alt: "the fraction 2 x over x minus 3")[$display(frac(2 x, x − 3))$] needs a factor of #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis")[$( x + 2 )$] in the denominator, so we multiply numerator and denominator by #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis")[$( x + 2 )$]:


          #math.equation(block: true, alt: "the fraction 2 x over x minus 3 equals the fraction 2 x open parenthesis x plus 2 close parenthesis over open parenthesis x minus 3 close parenthesis open parenthesis x plus 2 close parenthesis equals the fraction 2 x squared plus 4 x over x squared minus x minus 6")[$frac(2 x, x − 3) = frac(2 x ( bold(italic(x)) + 2 ), ( x − 3 ) ( bold(italic(x)) + 2 )) = frac(2 x^(2) + 4 x, x^(2) − x − 6)$]
]
]


      The two new fractions we obtained in Example are like fractions; they have the same denominator.


    
    
      
=== Sums and Differences of Unlike Fractions


      We are now ready to add or subtract algebraic fractions with unlike denominators. We will do this in four steps.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Add or Subtract Fractions with Unlike Denominators:]


        + Find the LCD for the given fractions.
+ Build each fraction to an equivalent fraction with the LCD as its denominator.
+ Add or subtract the numerators of the resulting like fractions. Use the LCD as the denominator of the sum or difference.
+ Reduce the sum or difference, if possible.
]


      #examplebox("Example 10")[][
Subtract #math.equation(block: false, alt: "the fraction 3 x over x plus 2 minus the fraction 2 x over x minus 3")[$display(frac(3 x, x + 2)) − display(frac(2 x, x − 3))$].


        
        #solutionbox[
+ The LCD for these fractions is #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x minus 3 close parenthesis")[$( x + 2 ) ( x − 3 )$].
+ We build each fraction to an equivalent one with the LCD, as we did in Example.
                #math.equation(block: true, alt: "the fraction 3 x over x plus 2 equals the fraction 3 x squared minus 9 x over x squared minus x minus 6 and the fraction 2 x over x minus 3 equals the fraction 2 x squared plus 4 x over x squared minus x minus 6")[$frac(3 x, x + 2) = frac(3 x^(2) − 9 x, x^(2) − x − 6) " " " " " and " " " " " frac(2 x, x − 3) = frac(2 x^(2) + 4 x, x^(2) − x − 6)$]
+ Combine the numerators over the same denominator.
                #math.equation(block: true, alt: "the fraction 3 x over x plus 2 minus the fraction 2 x over x minus 3, equals the fraction 3 x squared minus 9 x over x squared minus x minus 6 minus the fraction 2 x squared plus 4 x over x squared minus x minus 6, Subtract the numerators.; equals the fraction open parenthesis 3 x squared minus 9 x close parenthesis minus open parenthesis 2 x squared plus 4 x close parenthesis over x squared minus x minus 6; equals the fraction x squared minus 13 x over x squared minus x minus 6")[$frac(3 x, x + 2) − frac(2 x, x − 3) & = frac(3 x^(2) − 9 x, x^(2) − x − 6) − frac(2 x^(2) + 4 x, x^(2) − x − 6) &  & "Subtract the numerators." \  & = frac(( 3 x^(2) − 9 x ) − ( 2 x^(2) + 4 x ), x^(2) − x − 6) \  & = frac(x^(2) − 13 x, x^(2) − x − 6)$]
+ Reduce the result, if possible. If we factor both numerator and denominator, we find
                #math.equation(block: true, alt: "the fraction x open parenthesis x minus 13 close parenthesis over open parenthesis x minus 3 close parenthesis open parenthesis x plus 2 close parenthesis")[$frac(x ( x − 13 ), ( x − 3 ) ( x + 2 ))$]
                The fraction cannot be reduced.
]
]


      #examplebox("Example 11")[][
Write as a single fraction: #math.equation(block: false, alt: "1 plus the fraction 2 over a minus the fraction a squared plus 2 over a squared plus a")[$display(1 + frac(2, a) − frac(a^(2) + 2, a^(2) + a))$].


        
        #solutionbox[
+ To find the LCD, factor each denominator:
                #math.equation(block: true, alt: "a, equals a; a squared plus a, equals a open parenthesis a plus 1 close parenthesis")[$a & = a \ a^(2) + a & = a ( a + 1 )$]
                The LCD is #math.equation(block: false, alt: "a open parenthesis a plus 1 close parenthesis")[$a ( a + 1 )$].
+ Build each term to an equivalent fraction with the LCD as denominator. (The building factors for each fraction are shown in color.) The third fraction already has the LCD for its denominator.
                #math.equation(block: true, alt: "1 equals the fraction 1 times a open parenthesis a plus 1 close parenthesis over 1 times a open parenthesis a plus 1 close parenthesis, equals the fraction a squared plus a over a open parenthesis a plus 1 close parenthesis; the fraction 2 over a equals the fraction 2 times open parenthesis a plus 1 close parenthesis over a times open parenthesis a plus 1 close parenthesis, equals the fraction 2 a plus 2 over a open parenthesis a plus 1 close parenthesis; the fraction a squared plus 2 over a squared plus a, equals the fraction a squared plus 2 over a open parenthesis a plus 1 close parenthesis")[$1 = frac(1 ⋅ bold(italic(a)) ( bold(italic(a)) + 1 ), 1 ⋅ bold(italic(a)) ( bold(italic(a)) + 1 )) & = frac(a^(2) + a, a ( a + 1 )) \ frac(2, a) = frac(2 ⋅ ( bold(italic(a)) + 1 ), a ⋅ ( bold(italic(a)) + 1 )) & = frac(2 a + 2, a ( a + 1 )) \ frac(a^(2) + 2, a^(2) + a) & = frac(a^(2) + 2, a ( a + 1 ))$]
+ Combine the numerators over the LCD.
                #math.equation(block: true, alt: "1 plus the fraction 2 over a minus the fraction a squared plus 2 over a squared plus a, equals the fraction a squared plus a over a open parenthesis a plus 1 close parenthesis plus the fraction 2 a plus 2 over a open parenthesis a plus 1 close parenthesis minus the fraction a squared plus 2 over a open parenthesis a plus 1 close parenthesis; equals the fraction a squared plus a plus open parenthesis 2 a plus 2 close parenthesis minus open parenthesis a squared plus 2 close parenthesis over a open parenthesis a plus 1 close parenthesis; equals the fraction 3 a over a open parenthesis a plus 1 close parenthesis")[$1 + frac(2, a) − frac(a^(2) + 2, a^(2) + a) & = frac(a^(2) + a, a ( a + 1 )) + frac(2 a + 2, a ( a + 1 )) − frac(a^(2) + 2, a ( a + 1 )) \  & = frac(a^(2) + a + ( 2 a + 2 ) − ( a^(2) + 2 ), a ( a + 1 )) \  & = frac(3 a, a ( a + 1 ))$]
+ Reduce the fraction to find
                #math.equation(block: true, alt: "the fraction 3 a over a open parenthesis a plus 1 close parenthesis equals the fraction 3 over a plus 1")[$frac(3 cancel(a), cancel(a) ( a + 1 )) = frac(3, a + 1)$]
]
]


    
    
      
=== Complex Fractions


      A fraction that contains one or more fractions in either its numerator or its denominator or both is called a complex fraction. For example,


      #math.equation(block: true, alt: "the fraction the fraction 2 over 3 over the fraction 5 over 6 and the fraction x plus the fraction 3 over 4 over x minus the fraction 1 over 2")[$display(frac(display(frac(2, 3)), display(frac(5, 6)))) " " " " " " " and " " " " " " " display(frac(x + display(frac(3, 4)), x − display(frac(1, 2))))$]
      are complex fractions. Like simple fractions, complex fractions represent quotients. For the examples above,


      #math.equation(block: true, alt: "the fraction the fraction 2 over 3 over the fraction 5 over 6 equals the fraction 2 over 3 divided by the fraction 5 over 6 and the fraction x plus the fraction 3 over 4 over x minus the fraction 1 over 2 equals open parenthesis x plus the fraction 3 over 4 close parenthesis divided by open parenthesis x minus the fraction 1 over 2 close parenthesis")[$display(frac(display(frac(2, 3)), display(frac(5, 6)))) = display(frac(2, 3)) ÷ display(frac(5, 6)) " " " " " " " " " " " and " " " " " " " " " " " display(frac(x + display(frac(3, 4)), x − display(frac(1, 2)))) = ( x + display(frac(3, 4)) ) ÷ ( x − display(frac(1, 2)) )$]
      We can always simplify a complex fraction into a standard algebraic fraction. If the denominator of the complex fraction is a single term, we can treat the fraction as a division problem and multiply the numerator by the reciprocal of the denominator. Thus,


      #math.equation(block: true, alt: "the fraction the fraction 2 over 3 over the fraction 5 over 6 equals the fraction 2 over 3 divided by the fraction 5 over 6 equals the fraction 2 over 3 times the fraction 6 over 5 equals the fraction 4 over 5")[$display(frac(display(frac(2, 3)), display(frac(5, 6)))) = display(frac(2, 3)) ÷ display(frac(5, 6)) = display(frac(2, 3)) ⋅ display(frac(6, 5)) = display(frac(4, 5))$]
      If the numerator or denominator of the complex fraction contains more than one term, it is easier to use the fundamental principle of fractions to simplify the expression.


      #examplebox("Example 12")[][
Simplify #math.equation(block: false, alt: "the fraction x plus the fraction 3 over 4 over x minus the fraction 1 over 2")[$" " " " " " display(frac(x + display(frac(3, 4)), x − display(frac(1, 2))))$]


        
        #solutionbox[
Consider all of the simple fractions that appear in the complex fraction; in this example #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] and #math.equation(block: false, alt: "the fraction 3 over 4")[$display(frac(3, 4))$]. The LCD of these fractions is #math.equation(block: false, alt: "4")[$4$]. If we multiply the numerator and denominator of the complex fraction by #math.equation(block: false, alt: "4")[$4$], we will eliminate the fractions within the fraction.


          Be sure to multiply #emph[each] term of the numerator and #emph[each] term of the denominator by #math.equation(block: false, alt: "4")[$4$].


          #math.equation(block: true, alt: "the fraction 4 open parenthesis x plus the fraction 3 over 4 close parenthesis over 4 open parenthesis x minus the fraction 1 over 2 close parenthesis equals the fraction 4 open parenthesis x close parenthesis plus 4 open parenthesis the fraction 3 over 4 close parenthesis over 4 open parenthesis x close parenthesis minus 4 open parenthesis the fraction 1 over 2 close parenthesis equals the fraction 4 x plus 3 over 4 x minus 2")[$frac(4 ( x + display(frac(3, 4)) ), 4 ( x − display(frac(1, 2)) )) = frac(4 ( x ) + 4 ( display(frac(3, 4)) ), 4 ( x ) − 4 ( display(frac(1, 2)) )) = frac(4 x + 3, 4 x − 2)$]
          Thus, the original complex fraction is equivalent to the simple fraction #math.equation(block: false, alt: "the fraction 4 x plus 3 over 4 x minus 2")[$display(frac(4 x + 3, 4 x − 2))$].
]
]


      We summarize the method for simplifying complex fractions as follows.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Simplify a Complex Fraction:]


        + Find the LCD of all the fractions contained in the complex fraction.
+ Multiply each term in the numerator and the denominator of the complex fraction by the LCD.
+ Reduce the resulting simple fraction, if possible.
]


    
    
      
=== Negative Exponents


      Algebraic fractions are sometimes written using negative exponents. (You can review negative exponents in Variation.


      #examplebox("Example 13")[][
Write each expression as a single algebraic fraction.


          + #math.equation(block: false, alt: "x to the power minus 1 minus y to the power minus 1")[$x^(− 1) − y^(− 1)$]
+ #math.equation(block: false, alt: "open parenthesis x to the power minus 2 plus y to the power minus 2 close parenthesis to the power minus 1")[$attach(( x^(− 2) + y^(− 2) ), t: − 1)$]


        
        #solutionbox[
+ #math.equation(block: false, alt: "x to the power minus 1 minus y to the power minus 1 equals the fraction 1 over x minus the fraction 1 over y or the fraction y minus x over x y")[$display(x^(− 1) − y^(− 1) = frac(1, x) − frac(1, y) " " " " " " " or " " " " " " " frac(y − x, x y))$]
+ #math.equation(block: false, alt: "open parenthesis x to the power minus 2 plus y to the power minus 2 close parenthesis to the power minus 1 equals open parenthesis the fraction 1 over x squared plus the fraction 1 over y squared close parenthesis to the power minus 1 equals open parenthesis the fraction y squared plus x squared over x squared y squared close parenthesis to the power minus 1 equals the fraction x squared y squared over y squared plus x squared")[$display(attach(( x^(− 2) + y^(− 2) ), t: − 1) = attach(( frac(1, x^(2)) + frac(1, y^(2)) ), t: − 1) = attach(( frac(y^(2) + x^(2), x^(2) y^(2)) ), t: − 1) = frac(x^(2) y^(2), y^(2) + x^(2)))$]
]
]


      When working with fractions and exponents, it is important to avoid some tempting but #emph[incorrect] algebraic operations.


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ In Examplea, note that
                #math.equation(block: true, alt: "the fraction 1 over x minus the fraction 1 over y not equal to the fraction 1 over x minus y")[$frac(1, x) − frac(1, y) ≠ frac(1, x − y)$]
        For example, you can check that for #math.equation(block: false, alt: "x equals 2")[$x = 2$] and #math.equation(block: false, alt: "y equals 3")[$y = 3$],
                #math.equation(block: true, alt: "the fraction 1 over 2 minus the fraction 1 over 3 not equal to the fraction 1 over 2 minus 3 equals minus 1")[$frac(1, 2) − frac(1, 3) ≠ frac(1, 2 − 3) = − 1$]
+ In Exampleb, note that
                 #math.equation(block: true, alt: "open parenthesis x to the power minus 2 plus y to the power minus 2 close parenthesis to the power minus 1 not equal to x squared plus y squared")[$attach(( x^(− 2) + y^(− 2) ), t: − 1) ≠ x^(2) + y^(2)$]


        In general, the fourth law of exponents does #emph[not] apply to sums and differences; that is,


        #math.equation(block: true, alt: "open parenthesis a plus b close parenthesis to the power n not equal to a to the power n plus b to the power n")[$( a + b )^(n) ≠ a^(n) + b^(n)$]
]


    
    
      
=== Section Summary


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Rational expression
- Building factor
- Like fraction
- Reciprocal
- Common factor
- Algebraic fraction
- Complex fraction
- Common denominator
- Numerator
- Common term
- Reduce
- Polynomial division
- Denominator
- Opposite


      
      
        
==== SKILLS


        Practice each skill in the exercises listed.


        + Reduce fractions: \#1–24
+ Multiply fractions: \#25–36
+ Divide fractions: \#37–48
+ Add like fractions: \#49–56
+ Find the LCD: \#57–62
+ Add unlike fractions: \#63–82
+ Simplify complex fractions: \#83–106


      
    
    
      
=== Exercises A.9


      For Problems 1-20, reduce the algebraic fraction.


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


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


        
      
      
        #math.equation(block: true, alt: "the fraction minus 12 r squared s t over minus 6 r s t squared")[$display(frac(− 12 r^(2) s t, − 6 r s t^(2)))$]


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


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


        
      
      
        #math.equation(block: true, alt: "the fraction 2 y minus 8 over 8")[$display(frac(2 y − 8, 8))$]


      
      
        #math.equation(block: true, alt: "the fraction 6 a cubed minus 4 a squared over 4 a")[$display(frac(6 a^(3) − 4 a^(2), 4 a))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 3 x cubed minus 6 x squared over 6 x squared")[$display(frac(3 x^(3) − 6 x^(2), 6 x^(2)))$]


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


        
          #math.equation(block: true, alt: "the fraction 6 open parenthesis 1 plus t close parenthesis over 1 minus t")[$display(frac(6 ( 1 + t ), 1 − t))$]


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


      
      
        #math.equation(block: true, alt: "the fraction 2 y squared minus 8 over 2 y plus 4")[$display(frac(2 y^(2) − 8, 2 y + 4))$]


        
          #math.equation(block: true, alt: "y minus 2")[$y − 2$]


        
      
      
        #math.equation(block: true, alt: "the fraction 5 y squared minus 20 over 2 y minus 4")[$display(frac(5 y^(2) − 20, 2 y − 4))$]


      
      
        #math.equation(block: true, alt: "the fraction 6 minus 2 v over v cubed minus 27")[$display(frac(6 − 2 v, v^(3) − 27))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 4 minus 2 u over u cubed minus 8")[$display(frac(4 − 2 u, u^(3) − 8))$]


      
      
        #math.equation(block: true, alt: "the fraction 4 x cubed minus 36 x over 6 x squared plus 18 x")[$display(frac(4 x^(3) − 36 x, 6 x^(2) + 18 x))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 5 x squared plus 10 x over 5 x cubed plus 20 x")[$display(frac(5 x^(2) + 10 x, 5 x^(3) + 20 x))$]


      
      
        #math.equation(block: true, alt: "the fraction y squared minus 9 x squared over open parenthesis 3 x minus y close parenthesis squared")[$display(frac(y^(2) − 9 x^(2), ( 3 x − y )^(2)))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction open parenthesis 2 x minus y close parenthesis squared over y squared minus 4 x squared")[$display(frac(( 2 x − y )^(2), y^(2) − 4 x^(2)))$]


      
      
        #math.equation(block: true, alt: "the fraction 2 x squared plus x minus 6 over x squared plus x minus 2")[$display(frac(2 x^(2) + x − 6, x^(2) + x − 2))$]


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


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


      
      
        #math.equation(block: true, alt: "the fraction 8 z cubed minus 27 over 4 z squared minus 9")[$display(frac(8 z^(3) − 27, 4 z^(2) − 9))$]


        
          #math.equation(block: true, alt: "the fraction 4 z squared plus 6 z plus 9 over 2 z plus 3")[$display(frac(4 z^(2) + 6 z + 9, 2 z + 3))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 8 z cubed minus 1 over 4 z squared minus 1")[$display(frac(8 z^(3) − 1, 4 z^(2) − 1))$]


      
      
        Which of the following fractions are equivalent to #math.equation(block: false, alt: "2 a")[$2 a$] (on their common domain)?


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


        
          (b)


        
      
      
        Which of the following fractions are equivalent to #math.equation(block: false, alt: "3 b")[$3 b$] (on their common domain)?


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


      
      
        Which of the following fractions are equivalent to #math.equation(block: false, alt: "minus 1")[$− 1$] (where they are defined)?


        + #math.equation(block: false, alt: "the fraction 2 a plus b over 2 a minus b")[$display(frac(2 a + b, 2 a − b))$]
+ #math.equation(block: false, alt: "the fraction minus open parenthesis a plus b close parenthesis over b minus a")[$display(frac(− ( a + b ), b − a))$]
+ #math.equation(block: false, alt: "the fraction 2 a squared minus 1 over 2 a squared")[$display(frac(2 a^(2) − 1, 2 a^(2)))$]
+ #math.equation(block: false, alt: "the fraction minus a squared plus 3 over a squared plus 3")[$display(frac(− a^(2) + 3, a^(2) + 3))$]


        
          None


        
      
      
        Which of the following fractions are equivalent to #math.equation(block: false, alt: "minus 1")[$− 1$] (where they are defined)?


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


      
      For Problems 25-36, write the product as a single fraction in lowest terms.


      
        #math.equation(block: true, alt: "the fraction minus 4 over 3 n p times the fraction 6 n squared p cubed over 16")[$display(frac(− 4, 3 n p)) ⋅ display(frac(6 n^(2) p^(3), 16))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 14 a cubed b over 3 b times the fraction minus 6 over 7 a squared")[$display(frac(14 a^(3) b, 3 b)) ⋅ display(frac(− 6, 7 a^(2)))$]


      
      
        #math.equation(block: true, alt: "5 a squared b squared times the fraction 1 over a cubed b cubed")[$5 a^(2) b^(2) ⋅ display(frac(1, a^(3) b^(3)))$]


        
          #math.equation(block: true, alt: "the fraction 5 over a b")[$display(frac(5, a b))$]


        
      
      
        #math.equation(block: true, alt: "15 x squared y times the fraction 3 over 35 x y squared")[$15 x^(2) y ⋅ display(frac(3, 35 x y^(2)))$]


      
      
        #math.equation(block: true, alt: "the fraction 5 x plus 25 over 2 x times the fraction 4 x over 2 x plus 10")[$display(frac(5 x + 25, 2 x)) ⋅ display(frac(4 x, 2 x + 10))$]


        
          #math.equation(block: true, alt: "5")[$5$]


        
      
      
        #math.equation(block: true, alt: "the fraction 3 y over 4 x y minus 6 y squared times the fraction 2 x minus 3 y over 12 x")[$display(frac(3 y, 4 x y − 6 y^(2))) ⋅ display(frac(2 x − 3 y, 12 x))$]


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


        
          #math.equation(block: true, alt: "the fraction a open parenthesis 2 a minus 1 close parenthesis over a plus 4")[$display(frac(a ( 2 a − 1 ), a + 4))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 9 x squared minus 25 over 2 x minus 2 times the fraction x squared minus 1 over 6 x minus 10")[$display(frac(9 x^(2) − 25, 2 x − 2)) ⋅ display(frac(x^(2) − 1, 6 x − 10))$]


      
      
        #math.equation(block: true, alt: "the fraction 2 x squared minus x minus 6 over 3 x squared plus 4 x plus 1 times the fraction 3 x squared plus 7 x plus 2 over 2 x squared plus 7 x plus 6")[$display(frac(2 x^(2) − x − 6, 3 x^(2) + 4 x + 1)) ⋅ display(frac(3 x^(2) + 7 x + 2, 2 x^(2) + 7 x + 6))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 3 x squared minus 7 x minus 6 over 2 x squared minus x minus 1 times the fraction 2 x squared minus 9 x minus 5 over 3 x squared minus 13 x minus 10")[$display(frac(3 x^(2) − 7 x − 6, 2 x^(2) − x − 1)) ⋅ display(frac(2 x^(2) − 9 x − 5, 3 x^(2) − 13 x − 10))$]


      
      
        #math.equation(block: true, alt: "the fraction 3 x to the power 4 minus 48 over x to the power 4 minus 4 x squared minus 32 times the fraction 4 x to the power 4 minus 8 x cubed plus 4 x squared over 2 x to the power 4 plus 16 x")[$display(frac(3 x^(4) − 48, x^(4) − 4 x^(2) − 32)) ⋅ display(frac(4 x^(4) − 8 x^(3) + 4 x^(2), 2 x^(4) + 16 x))$]


        
          #math.equation(block: true, alt: "the fraction 6 x open parenthesis x minus 2 close parenthesis open parenthesis x minus 1 close parenthesis squared over open parenthesis x squared minus 8 close parenthesis open parenthesis x squared minus 2 x plus 4 close parenthesis")[$display(frac(6 x ( x − 2 ) ( x − 1 )^(2), ( x^(2) − 8 ) ( x^(2) − 2 x + 4 )))$]


        
      
      
        #math.equation(block: true, alt: "the fraction x to the power 4 minus 3 x cubed over x to the power 4 plus 6 x squared minus 27 times the fraction x to the power 4 minus 81 over 3 x to the power 4 minus 81 x")[$display(frac(x^(4) − 3 x^(3), x^(4) + 6 x^(2) − 27)) ⋅ display(frac(x^(4) − 81, 3 x^(4) − 81 x))$]


      
      For Problems 37-48, write the quotient as a single fraction in lowest terms.


      
        #math.equation(block: true, alt: "the fraction 4 x minus 8 over 3 y divided by the fraction 6 x minus 12 over y")[$display(frac(4 x − 8, 3 y)) ÷ display(frac(6 x − 12, y))$]


        
          #math.equation(block: true, alt: "the fraction 2 over 9")[$display(frac(2, 9))$]


        
      
      
        #math.equation(block: true, alt: "the fraction 6 y minus 27 over 5 x divided by the fraction 4 y minus 18 over x")[$display(frac(6 y − 27, 5 x)) ÷ display(frac(4 y − 18, x))$]


      
      
        #math.equation(block: true, alt: "the fraction a squared minus a minus 6 over a squared plus 2 a minus 15 divided by the fraction a squared minus 4 over a squared plus 6 a plus 5")[$display(frac(a^(2) − a − 6, a^(2) + 2 a − 15)) ÷ display(frac(a^(2) − 4, a^(2) + 6 a + 5))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction a squared plus 2 a minus 15 over a squared plus 3 a minus 10 divided by the fraction a squared minus 9 over a squared minus 9 a plus 14")[$display(frac(a^(2) + 2 a − 15, a^(2) + 3 a − 10)) ÷ display(frac(a^(2) − 9, a^(2) − 9 a + 14))$]


      
      
        #math.equation(block: true, alt: "the fraction x cubed plus y cubed over x divided by the fraction x plus y over 3 x")[$display(frac(x^(3) + y^(3), x)) ÷ display(frac(x + y, 3 x))$]


        
          #math.equation(block: true, alt: "3 open parenthesis x squared minus x y plus y squared close parenthesis")[$3 ( x^(2) − x y + y^(2) )$]


        
      
      
        #math.equation(block: true, alt: "the fraction 8 x cubed minus y cubed over x plus y divided by the fraction 2 x minus y over x squared minus y squared")[$display(frac(8 x^(3) − y^(3), x + y)) ÷ display(frac(2 x − y, x^(2) − y^(2)))$]


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


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


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


      
      
        #math.equation(block: true, alt: "open parenthesis x squared minus 5 x plus 4 close parenthesis divided by the fraction x squared minus 1 over x squared")[$( x^(2) − 5 x + 4 ) ÷ display(frac(x^(2) − 1, x^(2)))$]


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


        
      
      
        #math.equation(block: true, alt: "open parenthesis x squared minus 9 close parenthesis divided by the fraction x squared minus 6 x plus 9 over 3 x")[$( x^(2) − 9 ) ÷ display(frac(x^(2) − 6 x + 9, 3 x))$]


      
      
        #math.equation(block: true, alt: "the fraction x squared plus 3 x over 2 y divided by open parenthesis 3 x close parenthesis")[$display(frac(x^(2) + 3 x, 2 y)) ÷ ( 3 x )$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 2 y squared plus y over 3 x divided by open parenthesis 2 y close parenthesis")[$display(frac(2 y^(2) + y, 3 x)) ÷ ( 2 y )$]


      
      For Problems 49-56, write the sum or difference as a single fraction in lowest terms.


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


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


        
      
      
        #math.equation(block: true, alt: "the fraction y over 7 minus the fraction 5 over 7")[$display(frac(y, 7)) − display(frac(5, 7))$]


      
      
        #math.equation(block: true, alt: "the fraction 1 over 6 a plus the fraction 1 over 6 b minus the fraction 5 over 6 c")[$display(frac(1, 6)) a + display(frac(1, 6)) b − display(frac(5, 6)) c$]


        
          #math.equation(block: true, alt: "the fraction a plus b minus 5 c over 6")[$display(frac(a + b − 5 c, 6))$]


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


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


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


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


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


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


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


      
      For Problems 57-62, find the LCD for the pair of fractions.


      
        #math.equation(block: true, alt: "the fraction 5 over 6 open parenthesis x plus y close parenthesis squared , the fraction 3 over 4 x y squared")[$display(frac(5, 6 ( x + y )^(2))) " " , " " " " " " display(frac(3, 4 x y^(2)))$]


        
          #math.equation(block: true, alt: "12 x y squared open parenthesis x plus y close parenthesis squared")[$12 x y^(2) ( x + y )^(2)$]


        
      
      
        #math.equation(block: true, alt: "the fraction 1 over 8 open parenthesis a minus b close parenthesis squared , the fraction 5 over 12 a squared b squared")[$display(frac(1, 8 ( a − b )^(2))) " " , " " " " " " display(frac(5, 12 a^(2) b^(2)))$]


      
      
        #math.equation(block: true, alt: "the fraction 2 a over a squared plus 5 a plus 4 , the fraction 2 over open parenthesis a plus 1 close parenthesis squared")[$display(frac(2 a, a^(2) + 5 a + 4)) " " , " " " " " " display(frac(2, ( a + 1 )^(2)))$]


        
          #math.equation(block: true, alt: "open parenthesis a plus 4 close parenthesis open parenthesis a plus 1 close parenthesis squared")[$( a + 4 ) ( a + 1 )^(2)$]


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


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


        
          #math.equation(block: true, alt: "x open parenthesis x minus 1 close parenthesis cubed")[$x ( x − 1 )^(3)$]


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


      
      For Problems 63-82, write the sum or difference as a single fraction in lowest terms.


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


        
          #math.equation(block: true, alt: "the fraction 7 x over 6")[$display(frac(7 x, 6))$]


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


      
      
        #math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 3 over 4 y")[$display(frac(5, 6)) y − display(frac(3, 4)) y$]


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


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


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


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


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


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


        
          #math.equation(block: true, alt: "the fraction 8 x minus 5 over x open parenthesis x minus 1 close parenthesis")[$display(frac(8 x − 5, x ( x − 1 )))$]


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


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


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


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


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


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


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


      
      
        #math.equation(block: true, alt: "the fraction 7 over 5 x minus 10 minus the fraction 5 over 3 x minus 6")[$display(frac(7, 5 x − 10)) − display(frac(5, 3 x − 6))$]


        
          #math.equation(block: true, alt: "the fraction minus 4 over 15 open parenthesis x minus 2 close parenthesis")[$display(frac(− 4, 15 ( x − 2 )))$]


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


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


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


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


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


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


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


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


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


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


      
      For Problems 83-94, write the complex fraction as a simple fraction in lowest terms.


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


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


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


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


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


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


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


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


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


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


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


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


      
      
        #math.equation(block: true, alt: "the fraction L plus C over the fraction 1 over L plus the fraction 1 over C")[$display(frac(L + C, display(frac(1, L)) + display(frac(1, C))))$]


        
          #math.equation(block: true, alt: "L C")[$L C$]


        
      
      
        #math.equation(block: true, alt: "the fraction H minus T over the fraction H over T minus the fraction T over H")[$display(frac(H − T, display(frac(H, T)) − display(frac(T, H))))$]


      
      
        #math.equation(block: true, alt: "the fraction the fraction 4 over x squared minus the fraction 4 over z squared over the fraction 2 over z minus the fraction 2 over x")[$display(frac(display(frac(4, x^(2))) − display(frac(4, z^(2))), display(frac(2, z)) − display(frac(2, x))))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction the fraction 6 over b minus the fraction 6 over a over the fraction 3 over a squared minus the fraction 3 over b squared")[$display(frac(display(frac(6, b)) − display(frac(6, a)), display(frac(3, a^(2))) − display(frac(3, b^(2)))))$]


      
      For Problems 95-106, write the expression as a single algebraic fraction.


      
        #math.equation(block: true, alt: "x to the power minus 2 plus y to the power minus 2")[$x^(− 2) + y^(− 2)$]


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


        
      
      
        #math.equation(block: true, alt: "x to the power minus 2 minus y to the power minus 2")[$x^(− 2) − y^(− 2)$]


      
      
        #math.equation(block: true, alt: "2 w to the power minus 1 minus open parenthesis 2 w close parenthesis to the power minus 2")[$2 w^(− 1) − ( 2 w )^(− 2)$]


        
          #math.equation(block: true, alt: "the fraction 8 w minus 1 over 4 w squared")[$display(frac(8 w − 1, 4 w^(2)))$]


        
      
      
        #math.equation(block: true, alt: "3 w to the power minus 3 plus open parenthesis 3 w close parenthesis to the power minus 1")[$3 w^(− 3) + ( 3 w )^(− 1)$]


      
      
        #math.equation(block: true, alt: "a to the power minus 1 b minus a b to the power minus 1")[$a^(− 1) b − a b^(− 1)$]


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


        
      
      
        #math.equation(block: true, alt: "a minus b to the power minus 1 a minus b to the power minus 1")[$a − b^(− 1) a − b^(− 1)$]


      
      
        #math.equation(block: true, alt: "open parenthesis x to the power minus 1 plus y to the power minus 1 close parenthesis to the power minus 1")[$( x^(− 1) + y^(− 1) )^(− 1)$]


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


        
      
      
        #math.equation(block: true, alt: "open parenthesis 1 minus x y to the power minus 1 close parenthesis to the power minus 1")[$( 1 − x y^(− 1) )^(− 1)$]


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


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


        
      
      
        #math.equation(block: true, alt: "the fraction x to the power minus 1 minus y over x to the power minus 1")[$display(frac(x^(− 1) − y, x^(− 1)))$]


      
      
        #math.equation(block: true, alt: "the fraction a to the power minus 1 plus b to the power minus 1 over open parenthesis a b close parenthesis to the power minus 1")[$display(frac(a^(− 1) + b^(− 1), ( a b )^(− 1)))$]


        
          #math.equation(block: true, alt: "b plus a")[$b + a$]


        
      
      
        #math.equation(block: true, alt: "the fraction x over x to the power minus 2 minus y to the power minus 2")[$display(frac(x, x^(− 2) − y^(− 2)))$]