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== 10.6#h(0.6em)Laws of Exponents


    In this section, we review the rules for performing operations on powers.


    
      
=== Product of Powers


      Consider a product of two powers with the same base.


      #math.equation(block: true, alt: "open parenthesis a cubed close parenthesis open parenthesis a squared close parenthesis equals a a a times a a equals a to the power 5")[$( a^(3) ) ( a^(2) ) = a a a ⋅ a a = a^(5)$]
      because #math.equation(block: false, alt: "a")[$a$] occurs as a factor five times. The number of #math.equation(block: false, alt: "a")[$a$]'s in the product is the #math.equation(block: false, alt: "s u m")[$s u m$] of the number of  #math.equation(block: false, alt: "a")[$a$]'s in each factor.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[First Law of Exponents: Product of Powers]


        To multiply two powers with the same base, add the exponents and leave the base unchanged.


        #math.equation(block: true, alt: "a to the power m times a to the power n equals a to the power m plus n")[$a^(m) ⋅ a^(n) = a^(m + n)$]
]


      #examplebox("Example 1")[][
#figure(figph[figure showing powers], alt: "figure showing powers", caption: none)
]


      Here are some mistakes to avoid.


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ Note that we do not #emph[multiply] the exponents when simplifying a product. For example,
            #math.equation(block: true, alt: "b to the power 4 times b squared not equal to b to the power 8")[$b^(4) ⋅ b^(2) ≠ b^(8)$]
            You can check this with your calculator by choosing a value for #math.equation(block: false, alt: "b")[$b$], for instance, #math.equation(block: false, alt: "b equals 3")[$b = 3$]:
                #math.equation(block: true, alt: "3 to the power 4 times 3 squared not equal to 3 to the power 8")[$3^(4) ⋅ 3^(2) ≠ 3^(8)$]
+ In order to apply the first law of exponents, the bases must be the same. For example,
                #math.equation(block: true, alt: "2 cubed times 3 to the power 5 not equal to 6 to the power 8")[$2^(3) ⋅ 3^(5) ≠ 6^(8)$]
            (Check this on your calculator.)
+ We do not multiply the bases when simplifying a product. In Examplea, note that
                #math.equation(block: true, alt: "5 cubed times 5 to the power 4 not equal to 25 to the power 7")[$5^(3) ⋅ 5^(4) ≠ 25^(7)$]
+ Although we can simplify the product #math.equation(block: false, alt: "x squared x cubed")[$x^(2) x^(3)$] as #math.equation(block: false, alt: "x to the power 5")[$x^(5)$], we cannot simplify the sum #math.equation(block: false, alt: "x squared plus x cubed")[$x^(2) + x^(3)$], because #math.equation(block: false, alt: "x squared")[$x^(2)$] and #math.equation(block: false, alt: "x cubed")[$x^(3)$] are not like terms.
]


      #examplebox("Example 2")[][
Multiply #math.equation(block: false, alt: "open parenthesis minus 3 x to the power 4 z squared close parenthesis open parenthesis 5 x cubed z close parenthesis")[$( − 3 x^(4) z^(2) ) ( 5 x^(3) z )$].


        
        #solutionbox[
Rearrange the factors to group the numerical coefficients and the powers of each base. Apply the first law of exponents.


          #math.equation(block: true, alt: "open parenthesis minus 3 x to the power 4 z squared close parenthesis open parenthesis 5 x cubed z close parenthesis, equals open parenthesis minus 3 close parenthesis open parenthesis 5 close parenthesis x to the power 4 x cubed z squared z; equals minus 15 x to the power 7 z cubed")[$( − 3 x^(4) z^(2) ) ( 5 x^(3) z ) & = ( − 3 ) ( 5 ) x^(4) x^(3) z^(2) z \  & = − 15 x^(7) z^(3)$]
]
]


    
    
      
=== Quotients of Powers


      To reduce a fraction, we divide both numerator and denominator by any common factors.


      #math.equation(block: true, alt: "the fraction x to the power 7 over x to the power 4 equals the fraction x x x x x x x over x x x x equals the fraction x cubed over 1 equals x cubed")[$frac(x^(7), x^(4)) = frac(x x x cancel(x) cancel(x) cancel(x) cancel(x), cancel(x) cancel(x) cancel(x) cancel(x)) = frac(x^(3), 1) = x^(3)$]
      We can obtain the same result more quickly by #emph[subtracting] the exponent of the denominator from the exponent of the numerator.


      #math.equation(block: true, alt: "the fraction x to the power 7 over x to the power 4 equals x to the power 7 minus 4 equals x cubed")[$frac(x^(7), x^(4)) = x^(7 − 4) = x^(3)$]
      What if the larger power occurs in the denominator of the fraction?


      #math.equation(block: true, alt: "the fraction x to the power 4 over x to the power 7 equals the fraction x x x x over x x x x x x x equals the fraction 1 over x cubed")[$frac(x^(4), x^(7)) = frac(cancel(x) cancel(x) cancel(x) cancel(x), x x x cancel(x) cancel(x) cancel(x) cancel(x)) = frac(1, x^(3))$]
      In this case, we subtract the exponent of the numerator from the exponent of the denominator.


      #math.equation(block: true, alt: "the fraction x to the power 4 over x to the power 7 equals the fraction 1 over x to the power 7 minus 4 equals the fraction 1 over x cubed")[$frac(x^(4), x^(7)) = frac(1, x^(7 − 4)) = frac(1, x^(3))$]
      These examples suggest the following law.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Second Law of Exponents: Quotient of Powers]


        To divide two powers with the same base, subtract the smaller exponent from the larger one, keeping the same base.


        + If the larger exponent occurs in the numerator, put the power in the numerator.
            #math.equation(block: true, alt: "If m greater than n , then the fraction a to the power m over a to the power n equals a to the power m minus n open parenthesis a not equal to 0 close parenthesis")[$"If " m > n , " " " then " " " frac(a^(m), a^(n)) = a^(m − n) #hide($b l a n k$) ( a ≠ 0 )$]
+ If the larger exponent occurs in the denominator, put the power in the denominator.
            #math.equation(block: true, alt: "If m less than n , then the fraction a to the power m over a to the power n equals the fraction 1 over a to the power n minus m open parenthesis a not equal to 0 close parenthesis")[$"If " m < n , " " " then " " " frac(a^(m), a^(n)) = frac(1, a^(n − m)) #hide($b l a n k$) ( a ≠ 0 )$]
]


      #examplebox("Example 3")[][
+ #math.equation(block: false, alt: "the fraction 3 to the power 8 over 3 squared equals 3 to the power 8 minus 2 equals 3 to the power 6 Subtract exponents: 8 greater than 2.")[$display(frac(3^(8), 3^(2))) = 3^(8 − 2) = 3^(6) #hide($b l a n k b l a n k b l a$) "Subtract exponents: " " " 8 > 2.$]
+ #math.equation(block: false, alt: "the fraction w cubed over w to the power 6 equals the fraction 1 over w to the power 6 minus 3 equals the fraction 1 over w cubed Subtract exponents: 3 less than 6.")[$display(frac(w^(3), w^(6)) = frac(1, w^(6 − 3)) = frac(1, w^(3)) #hide($b l a n k b l a n k$) "Subtract exponents: " 3 < 6.)$]
]


      #examplebox("Example 4")[][
Divide #math.equation(block: false, alt: "the fraction 3 x squared y to the power 4 over 6 x cubed y")[$" " " " display(frac(3 x^(2) y^(4), 6 x^(3) y))$]


        
        #solutionbox[
Consider the numerical coefficients and the powers of each variable separately. Use the second law of exponents to simplify each quotient of powers.


          #math.equation(block: true, alt: "the fraction 3 x squared y to the power 4 over 6 x cubed y, equals the fraction 3 over 6 times the fraction x squared over x cubed times the fraction y to the power 4 over y, Subtract exponents.; equals the fraction 1 over 2 times the fraction 1 over x to the power 3 minus 2 times y to the power 4 minus 1; equals the fraction 1 over 2 times the fraction 1 over x times y cubed equals the fraction y cubed over 2 x")[$frac(3 x^(2) y^(4), 6 x^(3) y) & = frac(3, 6) ⋅ frac(x^(2), x^(3)) ⋅ frac(y^(4), y) &  & "Subtract exponents." \  & = frac(1, 2) ⋅ frac(1, x^(3 − 2)) ⋅ y^(4 − 1) \  & = frac(1, 2) ⋅ frac(1, x) ⋅ y^(3) = frac(y^(3), 2 x)$]
]
]


    
    
      
=== Power of a Power


      Consider the expression #math.equation(block: false, alt: "open parenthesis a to the power 4 close parenthesis cubed")[$attach(( a^(4) ), t: 3)$], the third power of #math.equation(block: false, alt: "a to the power 4")[$a^(4)$].


      #math.equation(block: true, alt: "open parenthesis a to the power 4 close parenthesis cubed equals open parenthesis a to the power 4 close parenthesis open parenthesis a to the power 4 close parenthesis open parenthesis a to the power 4 close parenthesis equals a to the power 4 plus 4 plus 4 equals a to the power 12 Add exponents.")[$attach(( a^(4) ), t: 3) = ( a^(4) ) ( a^(4) ) ( a^(4) ) = a^(4 + 4 + 4) = a^(12) #hide($b l a n k b l a n k$) "Add exponents."$]
      We can obtain the same result by multiplying the exponents together.


      #math.equation(block: true, alt: "open parenthesis a to the power 4 close parenthesis cubed equals a to the power 4 times 3 equals a to the power 12")[$attach(( a^(4) ), t: 3) = a^(4 #h(0.167em) ⋅ #h(0.167em) 3) = a^(12)$]
      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Third Law of Exponents: Power of a Power]


        To raise a power to a power, keep the same base and multiply the exponents.


        #math.equation(block: true, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m n")[$attach(( a^(m) ), t: n) = a^(m n)$]
]


      #examplebox("Example 5")[][
#figure(figph[multiplying exponents], alt: "multiplying exponents", caption: none)
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Notice the difference between the expressions


        #math.equation(block: true, alt: "open parenthesis x cubed close parenthesis open parenthesis x to the power 4 close parenthesis equals x to the power 3 plus 4 equals x to the power 7")[$( x^(3) ) ( x^(4) ) = x^(3 + 4) = x^(7)$]
        and


        #math.equation(block: true, alt: "open parenthesis x cubed close parenthesis to the power 4 equals x to the power 3 times 4 equals x to the power 12")[$attach(( x^(3) ), t: 4) = x^(3 #h(0.167em) ⋅ #h(0.167em) 4) = x^(12)$]
        The first expression is a product, so we add the exponents. The second expression raises a
power to a power, so we multiply the exponents.
]


    
    
      
=== Power of a Product


      To simplify the expression #math.equation(block: false, alt: "open parenthesis 5 a close parenthesis cubed")[$( 5 a )^(3)$], we use the associative and commutative laws to regroup the factors as follows.


      #math.equation(block: true, alt: "open parenthesis 5 a close parenthesis cubed, equals open parenthesis 5 a close parenthesis open parenthesis 5 a close parenthesis open parenthesis 5 a close parenthesis; equals 5 times 5 times 5 times a times a times a; equals 5 cubed a cubed")[$( 5 a )^(3) & = ( 5 a ) ( 5 a ) ( 5 a ) \  & = 5 ⋅ 5 ⋅ 5 ⋅ a ⋅ a ⋅ a \  & = 5^(3) a^(3)$]
      Thus, to raise a product to a power, we can simply raise each factor to the power.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fourth Law of Exponents: Power of a Product]


        A power of a product is equal to the product of the powers of each of its factors.


        #math.equation(block: true, alt: "open parenthesis a b close parenthesis to the power n equals a to the power n b to the power n")[$( a b )^(n) = a^(n) b^(n)$]
]


      #examplebox("Example 6")[][
+ #math.equation(block: false, alt: "open parenthesis 5 a close parenthesis cubed equals 5 cubed a cubed equals 125 a cubed Cube each factor")[$( 5 a )^(3) = 5^(3) a^(3) = 125 a^(3) #hide($b l a n k$) "Cube each factor"$].
+ #math.equation(block: false, alt: "open parenthesis minus x y squared close parenthesis to the power 4, equals open parenthesis minus x close parenthesis to the power 4 open parenthesis y squared close parenthesis to the power 4, Raise each factor to the fourth power.; equals x to the power 4 y to the power 8, Apply the third law of exponents.")[$attach(( − x y^(2) ), t: 4) & = ( − x )^(4) attach(( y^(2) ), t: 4) &  & "Raise each factor to the fourth power." \  & = x^(4) y^(8) &  & "Apply the third law of exponents."$]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ Compare the two expressions #math.equation(block: false, alt: "3 a squared")[$3 a^(2)$] and #math.equation(block: false, alt: "open parenthesis 3 a close parenthesis squared")[$( 3 a )^(2)$]; they are not the same. In the expression #math.equation(block: false, alt: "3 a squared")[$3 a^(2)$], only the factor #math.equation(block: false, alt: "a")[$a$] is squared. But in #math.equation(block: false, alt: "open parenthesis 3 a close parenthesis squared")[$( 3 a )^(2)$], both #math.equation(block: false, alt: "3")[$3$] and #math.equation(block: false, alt: "a")[$a$] are squared. Thus,
        #math.equation(block: true, alt: "3 a squared cannot be simplified")[$3 a^(2) " " " " " cannot be simplified"$]
    but
        #math.equation(block: true, alt: "open parenthesis 3 a close parenthesis squared equals 3 squared a squared equals 9 a squared")[$( 3 a )^(2) = 3^(2) a^(2) = 9 a^(2)$]
+ Compare the two expressions #math.equation(block: false, alt: "open parenthesis 3 a close parenthesis squared")[$( 3 a )^(2)$] and #math.equation(block: false, alt: "open parenthesis 3 plus a close parenthesis squared")[$( 3 + a )^(2)$]. The fourth law of exponents applies to the #emph[product] #math.equation(block: false, alt: "3 a")[$3 a$], but not to the #emph[sum] #math.equation(block: false, alt: "3 plus a")[$3 + a$]. Thus,
    #math.equation(block: true, alt: "open parenthesis 3 plus a close parenthesis squared not equal to 3 squared plus a squared")[$( 3 + a )^(2) ≠ 3^(2) + a^(2)$]
    In order to simplify #math.equation(block: false, alt: "open parenthesis 3 plus a close parenthesis squared")[$( 3 + a )^(2)$], we must expand the binomial product:
    #math.equation(block: true, alt: "open parenthesis 3 plus a close parenthesis squared equals open parenthesis 3 plus a close parenthesis open parenthesis 3 plus a close parenthesis equals 9 plus 6 a plus a squared")[$( 3 + a )^(2) = ( 3 + a ) ( 3 + a ) = 9 + 6 a + a^(2)$]
]


    
    
      
=== Power of a Quotient


      To simplify the expression #math.equation(block: false, alt: "open parenthesis the fraction x over 3 close parenthesis to the power 4")[$display(attach(( frac(x, 3) ), t: 4))$], we multiply together #math.equation(block: false, alt: "4")[$4$] copies of the fraction 
    #math.equation(block: false, alt: "the fraction x over 3")[$display(frac(x, 3))$].


      #math.equation(block: true, alt: "open parenthesis the fraction x over 3 close parenthesis to the power 4, equals the fraction x over 3 times the fraction x over 3 times the fraction x over 3 times the fraction x over 3 equals the fraction x times x times x times x over 3 times 3 times 3 times 3; equals the fraction x to the power 4 over 3 to the power 4 equals the fraction x to the power 4 over 81")[$attach(( frac(x, 3) ), t: 4) & = frac(x, 3) ⋅ frac(x, 3) ⋅ frac(x, 3) ⋅ frac(x, 3) = frac(x ⋅ x ⋅ x ⋅ x, 3 ⋅ 3 ⋅ 3 ⋅ 3) \  & = frac(x^(4), 3^(4)) = frac(x^(4), 81)$]
      In general, we have the following rule.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fifth Law of Exponents: Power of a Quotient]


        To raise a quotient to a power, raise both the numerator and denominator to the power.


        #math.equation(block: true, alt: "open parenthesis the fraction a over b close parenthesis to the power n equals the fraction a to the power n over b to the power n")[$attach(( frac(a, b) ), t: n) = frac(a^(n), b^(n))$]
]


      For reference, we state all of the laws of exponents together. All the laws are valid when a and b are not equal to zero and when the exponents m and n are whole numbers.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Laws of Exponents]


        + #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n")[$a^(m) ⋅ a^(n) = a^(m + n)$]
+ + #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n m greater than n")[$display(frac(a^(m), a^(n))) = a^(m − n) #hide($b l a n k$) m > n$]
+ #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals the fraction 1 over a to the power n minus m m less than n")[$display(frac(a^(m), a^(n)) = frac(1, a^(n − m)) #hide($b l a n k$) m < n)$]
+ #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m plus n")[$attach(( a^(m) ), t: n) = a^(m + n)$]
+ #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power n equals a to the power n b to the power n")[$( a b )^(n) = a^(n) b^(n)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power n equals the fraction a to the power n over b to the power n")[$display(attach(( frac(a, b) ), t: n) = frac(a^(n), b^(n)))$]
]


      #examplebox("Example 7")[][
Simplify #math.equation(block: false, alt: "5 x squared y cubed open parenthesis 2 x y squared close parenthesis to the power 4")[$" " " " 5 x^(2) y^(3) attach(( 2 x y^(2) ), t: 4)$]


        
        #solutionbox[
According to the order of operations, we should perform any powers before multiplications. Thus, we begin by simplifying #math.equation(block: false, alt: "open parenthesis 2 x y squared close parenthesis to the power 4")[$( 2 x y^(2) )^(4)$]. We apply the fourth law.


          #math.equation(block: true, alt: "5 x squared y cubed open parenthesis 2 x y squared close parenthesis to the power 4, equals 5 x squared y cubed times 2 to the power 4 x to the power 4 open parenthesis y squared close parenthesis to the power 4, Apply the fourth law.; equals 5 x squared y cubed times 2 to the power 4 x to the power 4 y to the power 8")[$5 x^(2) y^(3) attach(( 2 x y^(2) ), t: 4) & = 5 x^(2) y^(3) ⋅ 2^(4) x^(4) attach(( y^(2) ), t: 4) &  & "Apply the fourth law." \  & = 5 x^(2) y^(3) ⋅ 2^(4) x^(4) y^(8)$]
          Finally, multiply powers with the same base. Apply the first law.


          #math.equation(block: true, alt: "5 x squared y cubed times 2 to the power 4 x to the power 4 y to the power 8 equals 5 times 2 to the power 4 x squared x to the power 4 y cubed y to the power 8 equals 80 x to the power 6 y to the power 11")[$5 x^(2) y^(3) ⋅ 2^(4) x^(4) y^(8) = 5 ⋅ 2^(4) x^(2) x^(4) y^(3) y^(8) = 80 x^(6) y^(11)$]
]
]


      #examplebox("Example 8")[][
Simplify #math.equation(block: false, alt: "open parenthesis the fraction 2 x over z squared close parenthesis cubed")[$" " " " display(attach(( frac(2 x, z^(2)) ), t: 3))$]


        
        #solutionbox[
Begin by applying the fifth law.


          #math.equation(block: true, alt: "open parenthesis the fraction 2 x over z squared close parenthesis cubed, equals the fraction open parenthesis 2 x close parenthesis cubed over open parenthesis z squared close parenthesis squared, Apply the fourth law to the numerator and the third law to the denominator.; equals the fraction 2 cubed x cubed over z to the power 6 equals the fraction 8 x cubed over z to the power 6")[$attach(( frac(2 x, z^(2)) ), t: 3) & = frac(( 2 x )^(3), attach(( z^(2) ), t: 2)) &  & "Apply the fourth law to the numerator and the third law to the denominator." \  & = frac(2^(3) x^(3), z^(6)) = frac(8 x^(3), z^(6))$]
]
]


    
    
      
=== Section Summary


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Exponent
- Power


      
      
        
==== SKILLS


        Practice each skill in the exercises listed.


        + Apply the laws of exponents: \#1–8
+ Simplify expressions: \#9–16, 25–32
+ Multiply and divide power: \#17–24


      
    
    
      
=== Exercises A.6


      For Problems 1–8, simplify by applying the appropriate law of exponents.


      
        + #math.equation(block: false, alt: "b to the power 4 times b to the power 5")[$b^(4) ⋅ b^(5)$]
+ #math.equation(block: false, alt: "b squared times b to the power 8")[$b^(2) ⋅ b^(8)$]
+ #math.equation(block: false, alt: "open parenthesis q cubed close parenthesis open parenthesis q close parenthesis open parenthesis q to the power 5 close parenthesis")[$( q^(3) ) ( q ) ( q^(5) )$]
+ #math.equation(block: false, alt: "open parenthesis p squared close parenthesis open parenthesis p to the power 4 close parenthesis open parenthesis p to the power 4 close parenthesis")[$( p^(2) ) ( p^(4) ) ( p^(4) )$]


        
          + #math.equation(block: false, alt: "b to the power 9")[$b^(9)$]
+ #math.equation(block: false, alt: "b to the power 10")[$b^(10)$]
+ #math.equation(block: false, alt: "q to the power 9")[$q^(9)$]
+ #math.equation(block: false, alt: "p to the power 10")[$p^(10)$]


        
      
      
        + #math.equation(block: false, alt: "the fraction w to the power 6 over w cubed")[$display(frac(w^(6), w^(3)))$]
+ #math.equation(block: false, alt: "the fraction c to the power 12 over c to the power 4")[$display(frac(c^(12), c^(4)))$]
+ #math.equation(block: false, alt: "the fraction z to the power 6 over z to the power 9")[$display(frac(z^(6), z^(9)))$]
+ #math.equation(block: false, alt: "the fraction b to the power 4 over b to the power 8")[$display(frac(b^(4), b^(8)))$]


      
      
        + #math.equation(block: false, alt: "2 to the power 7 times 2 squared")[$2^(7) ⋅ 2^(2)$]
+ #math.equation(block: false, alt: "6 to the power 5 times 6 cubed")[$6^(5) ⋅ 6^(3)$]
+ #math.equation(block: false, alt: "the fraction 2 to the power 9 over 2 to the power 4")[$display(frac(2^(9), 2^(4)))$]
+ #math.equation(block: false, alt: "the fraction 8 to the power 6 over 8 squared")[$display(frac(8^(6), 8^(2)))$]


        
          + #math.equation(block: false, alt: "2 to the power 9")[$2^(9)$]
+ #math.equation(block: false, alt: "6 to the power 8")[$6^(8)$]
+ #math.equation(block: false, alt: "2 to the power 5")[$2^(5)$]
+ #math.equation(block: false, alt: "8 to the power 4")[$8^(4)$]


        
      
      
        + #math.equation(block: false, alt: "open parenthesis d cubed close parenthesis to the power 5")[$( d^(3) )^(5)$]
+ #math.equation(block: false, alt: "open parenthesis d to the power 4 close parenthesis squared")[$( d^(4) )^(2)$]
+ #math.equation(block: false, alt: "open parenthesis 5 to the power 4 close parenthesis cubed")[$( 5^(4) )^(3)$]
+ #math.equation(block: false, alt: "open parenthesis 4 cubed close parenthesis cubed")[$( 4^(3) )^(3)$]


      
      
        + #math.equation(block: false, alt: "open parenthesis 6 x close parenthesis cubed")[$( 6 x )^(3)$]
+ #math.equation(block: false, alt: "open parenthesis 3 y close parenthesis to the power 4")[$( 3 y )^(4)$]
+ #math.equation(block: false, alt: "open parenthesis 2 t cubed close parenthesis to the power 5")[$( 2 t^(3) )^(5)$]
+ #math.equation(block: false, alt: "open parenthesis 6 s squared close parenthesis squared")[$( 6 s^(2) )^(2)$]


        
          + #math.equation(block: false, alt: "216 x cubed")[$216 x^(3)$]
+ #math.equation(block: false, alt: "81 y to the power 4")[$81 y^(4)$]
+ #math.equation(block: false, alt: "32 t to the power 15")[$32 t^(15)$]
+ #math.equation(block: false, alt: "36 s to the power 4")[$36 s^(4)$]


        
      
      
        + #math.equation(block: false, alt: "open parenthesis the fraction w over 2 close parenthesis to the power 6")[$attach(( display(frac(w, 2)) ), t: 6)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction 5 over u close parenthesis to the power 4")[$attach(( display(frac(5, u)) ), t: 4)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction minus 4 over p to the power 5 close parenthesis cubed")[$attach(( display(frac(− 4, p^(5))) ), t: 3)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction minus 3 over q to the power 4 close parenthesis to the power 5")[$attach(( display(frac(− 3, q^(4))) ), t: 5)$]


      
      
        + #math.equation(block: false, alt: "open parenthesis the fraction h squared over m cubed close parenthesis to the power 4")[$attach(( display(frac(h^(2), m^(3))) ), t: 4)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction n cubed over k to the power 4 close parenthesis to the power 8")[$attach(( display(frac(n^(3), k^(4))) ), t: 8)$]
+ #math.equation(block: false, alt: "open parenthesis minus 4 a squared b to the power 4 close parenthesis to the power 4")[$( − 4 a^(2) b^(4) )^(4)$]
+ #math.equation(block: false, alt: "open parenthesis minus 5 a b to the power 8 close parenthesis cubed")[$( − 5 a b^(8) )^(3)$]


        
          + #math.equation(block: false, alt: "the fraction h to the power 8 over m to the power 12")[$display(frac(h^(8), m^(12)))$]
+ #math.equation(block: false, alt: "the fraction n to the power 24 over k to the power 32")[$display(frac(n^(24), k^(32)))$]
+ #math.equation(block: false, alt: "2565 a to the power 8 b to the power 16")[$2565 a^(8) b^(16)$]
+ #math.equation(block: false, alt: "minus 125 a cubed b to the power 24")[$− 125 a^(3) b^(24)$]


        
      
      
        + #math.equation(block: false, alt: "the fraction a b squared over open parenthesis a b close parenthesis squared")[$display(frac(a b^(2), ( a b )^(2)))$]
+ #math.equation(block: false, alt: "the fraction open parenthesis x squared y close parenthesis squared over x squared y squared")[$display(frac(( x^(2) y )^(2), x^(2) y^(2)))$]
+ #math.equation(block: false, alt: "the fraction open parenthesis 2 m p close parenthesis cubed over 2 m cubed p")[$display(frac(( 2 m p )^(3), 2 m^(3) p))$]
+ #math.equation(block: false, alt: "the fraction 4 squared r t to the power 4 over 2 to the power 4 r to the power 4 t")[$display(frac(4^(2) r t^(4), 2^(4) r^(4) t))$]


      
      For Problems 9–15, simplify if possible.


      
        + #math.equation(block: false, alt: "w plus w")[$w + w$]
+ #math.equation(block: false, alt: "w open parenthesis w close parenthesis")[$w ( w )$]


        
          + #math.equation(block: false, alt: "2 w")[$2 w$]
+ #math.equation(block: false, alt: "w squared")[$w^(2)$]


        
      
      
        + #math.equation(block: false, alt: "m squared minus m squared")[$m^(2) − m^(2)$]
+ #math.equation(block: false, alt: "m squared open parenthesis minus m squared close parenthesis")[$m^(2) ( − m^(2) )$]


      
      
        + #math.equation(block: false, alt: "4 z squared minus 6 z squared")[$4 z^(2) − 6 z^(2)$]
+ #math.equation(block: false, alt: "4 z squared open parenthesis minus 6 z squared close parenthesis")[$4 z^(2) ( − 6 z^(2) )$]


        
          + #math.equation(block: false, alt: "minus 2 z squared")[$− 2 z^(2)$]
+ #math.equation(block: false, alt: "m minus 24 z to the power 4")[$m − 24 z^(4)$]


        
      
      
        + #math.equation(block: false, alt: "t cubed plus 3 t cubed")[$t^(3) + 3 t^(3)$]
+ #math.equation(block: false, alt: "t cubed open parenthesis 3 t cubed close parenthesis")[$t^(3) ( 3 t^(3) )$]


      
      
        + #math.equation(block: false, alt: "4 p squared plus 3 p cubed")[$4 p^(2) + 3 p^(3)$]
+ #math.equation(block: false, alt: "4 p squared open parenthesis 3 p cubed close parenthesis")[$4 p^(2) ( 3 p^(3) )$]


        
          + Cannot be simplified
+ #math.equation(block: false, alt: "12 p to the power 5")[$12 p^(5)$]


        
      
      
        + #math.equation(block: false, alt: "2 w squared minus 5 w to the power 4")[$2 w^(2) − 5 w^(4)$]
+ #math.equation(block: false, alt: "open parenthesis 2 w squared close parenthesis open parenthesis minus 5 w to the power 4 close parenthesis")[$( 2 w^(2) ) ( − 5 w^(4) )$]


      
      
        + #math.equation(block: false, alt: "3 to the power 9 times 3 to the power 8")[$3^(9) ⋅ 3^(8)$]
+ #math.equation(block: false, alt: "3 to the power 9 plus 3 to the power 8")[$3^(9) + 3^(8)$]


        
          + #math.equation(block: false, alt: "3 to the power 17")[$3^(17)$]
+ Cannot be simplified


        
      
      
        + #math.equation(block: false, alt: "open parenthesis minus 2 close parenthesis to the power 7 open parenthesis minus 2 close parenthesis to the power 5")[$( − 2 )^(7) ( − 2 )^(5)$]
+ #math.equation(block: false, alt: "minus 2 to the power 7 minus 2 to the power 5")[$− 2^(7) − 2^(5)$]


      
      F mor Problems 17–20, multiply.


      
        + #math.equation(block: false, alt: "open parenthesis 4 y close parenthesis open parenthesis minus 6 y close parenthesis")[$( 4 y ) ( − 6 y )$]
+ #math.equation(block: false, alt: "open parenthesis minus 4 z close parenthesis open parenthesis minus 8 z close parenthesis")[$( − 4 z ) ( − 8 z )$]


        
          + #math.equation(block: false, alt: "minus 24 y squared")[$− 24 y^(2)$]
+ #math.equation(block: false, alt: "32 z squared")[$32 z^(2)$]


        
      
      
        + #math.equation(block: false, alt: "open parenthesis 2 w z cubed close parenthesis open parenthesis minus 8 z close parenthesis")[$( 2 w z^(3) ) ( − 8 z )$]
+ #math.equation(block: false, alt: "open parenthesis 4 w z close parenthesis open parenthesis minus 9 w squared z squared close parenthesis")[$( 4 w z ) ( − 9 w^(2) z^(2) )$]


      
      
        + #math.equation(block: false, alt: "minus 4 x open parenthesis 3 x y close parenthesis open parenthesis x y cubed close parenthesis")[$− 4 x ( 3 x y ) ( x y^(3) )$]
+ #math.equation(block: false, alt: "open parenthesis minus 5 x squared close parenthesis open parenthesis 2 x y close parenthesis open parenthesis 5 x squared close parenthesis")[$( − 5 x^(2) ) ( 2 x y ) ( 5 x^(2) )$]


        
          + #math.equation(block: false, alt: "minus 12 x cubed y to the power 4")[$− 12 x^(3) y^(4)$]
+ #math.equation(block: false, alt: "minus 50 x to the power 5 y")[$− 50 x^(5) y$]


        
      
      
        + #math.equation(block: false, alt: "minus 7 a b squared open parenthesis minus 3 a b cubed close parenthesis")[$− 7 a b^(2) ( − 3 a b^(3) )$]
+ #math.equation(block: false, alt: "minus 4 a squared b open parenthesis minus 3 a cubed b squared close parenthesis")[$− 4 a^(2) b ( − 3 a^(3) b^(2) )$]


      
      F dor Problems 21–22, divide.


      
        + #math.equation(block: false, alt: "the fraction 2 a cubed b over 8 a to the power 4 b to the power 5")[$display(frac(2 a^(3) b, 8 a^(4) b^(5)))$]
+ #math.equation(block: false, alt: "the fraction 8 a squared b over 12 a to the power 5 b cubed")[$display(frac(8 a^(2) b, 12 a^(5) b^(3)))$]


        
          + #math.equation(block: false, alt: "the fraction 1 over 4 a b to the power 4")[$display(frac(1, 4 a b^(4)))$]
+ #math.equation(block: false, alt: "the fraction 2 over 3 a cubed b squared")[$display(frac(2, 3 a^(3) b^(2)))$]


        
      
      
        + #math.equation(block: false, alt: "the fraction minus 12 q w to the power 4 over 8 q w squared")[$display(frac(− 12 q w^(4), 8 q w^(2)))$]
+ #math.equation(block: false, alt: "the fraction minus 12 r z to the power 6 over 20 r z")[$display(frac(− 12 r z^(6), 20 r z))$]


      
      For Problems 23–24, multiply or divide.


      
        + #math.equation(block: false, alt: "the fraction minus 15 b c open parenthesis b squared c close parenthesis over minus 3 b cubed c to the power 4")[$display(frac(− 15 b c ( b^(2) c ), − 3 b^(3) c^(4)))$]
+ #math.equation(block: false, alt: "the fraction minus 25 c open parenthesis c squared d squared close parenthesis over minus 5 c to the power 8 d squared")[$display(frac(− 25 c ( c^(2) d^(2) ), − 5 c^(8) d^(2)))$]


        
          + #math.equation(block: false, alt: "the fraction 5 over c squared")[$display(frac(5, c^(2)))$]
+ #math.equation(block: false, alt: "the fraction 5 over c to the power 5")[$display(frac(5, c^(5)))$]


        
      
      
        + #math.equation(block: false, alt: "minus 2 x cubed open parenthesis x squared y close parenthesis open parenthesis minus 4 y squared close parenthesis")[$− 2 x^(3) ( x^(2) y ) ( − 4 y^(2) )$]
+ #math.equation(block: false, alt: "3 x y cubed open parenthesis minus x to the power 4 close parenthesis open parenthesis minus 2 y squared close parenthesis")[$3 x y^(3) ( − x^(4) ) ( − 2 y^(2) )$]


      
      For Problems 25–28, simplify by applying the laws of exponents.


      
        + #math.equation(block: false, alt: "b cubed open parenthesis b squared close parenthesis to the power 5")[$b^(3) ( b^(2) )^(5)$]
+ #math.equation(block: false, alt: "b open parenthesis b to the power 4 close parenthesis to the power 6")[$b ( b^(4) )^(6)$]


        
          + #math.equation(block: false, alt: "b to the power 13")[$b^(13)$]
+ #math.equation(block: false, alt: "b to the power 25")[$b^(25)$]


        
      
      
        + #math.equation(block: false, alt: "open parenthesis p squared q close parenthesis cubed open parenthesis p q cubed close parenthesis")[$( p^(2) q )^(3) ( p q^(3) )$]
+ #math.equation(block: false, alt: "open parenthesis p cubed close parenthesis to the power 4 open parenthesis p cubed q to the power 4 close parenthesis")[$( p^(3) )^(4) ( p^(3) q^(4) )$]


      
      
        + #math.equation(block: false, alt: "open parenthesis 2 x cubed y close parenthesis squared open parenthesis x y cubed close parenthesis to the power 4")[$( 2 x^(3) y )^(2) ( x y^(3) )^(4)$]
+ #math.equation(block: false, alt: "open parenthesis 3 x y squared close parenthesis cubed open parenthesis 2 x squared y squared close parenthesis squared")[$( 3 x y^(2) )^(3) ( 2 x^(2) y^(2) )^(2)$]


        
          + #math.equation(block: false, alt: "4 x to the power 10 y to the power 14")[$4 x^(10) y^(14)$]
+ #math.equation(block: false, alt: "108 x to the power 7 y to the power 10")[$108 x^(7) y^(10)$]


        
      
      
        + #math.equation(block: false, alt: "minus a squared open parenthesis minus a close parenthesis squared")[$− a^(2) ( − a )^(2)$]
+ #math.equation(block: false, alt: "minus a cubed open parenthesis minus a close parenthesis cubed")[$− a^(3) ( − a )^(3)$]


      
      For Problems 29–32, simplify by applying the laws of exponents.


      
        + #math.equation(block: false, alt: "open parenthesis the fraction minus 2 x over 3 y squared close parenthesis cubed")[$attach(( display(frac(− 2 x, 3 y^(2))) ), t: 3)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction minus x squared over 2 y close parenthesis to the power 4")[$attach(( display(frac(− x^(2), 2 y)) ), t: 4)$]


        
          + #math.equation(block: false, alt: "the fraction minus 8 x cubed over 27 y to the power 6")[$display(frac(− 8 x^(3), 27 y^(6)))$]
+ #math.equation(block: false, alt: "the fraction x to the power 8 over 16 y to the power 4")[$display(frac(x^(8), 16 y^(4)))$]


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


      
      
        + #math.equation(block: false, alt: "the fraction open parenthesis x y close parenthesis squared open parenthesis minus x squared y close parenthesis cubed over open parenthesis x squared y squared close parenthesis squared")[$display(frac(( x y )^(2) ( − x^(2) y )^(3), ( x^(2) y^(2) )^(2)))$]
+ #math.equation(block: false, alt: "the fraction open parenthesis minus x squared close parenthesis open parenthesis minus x squared close parenthesis to the power 4 over open parenthesis x squared close parenthesis cubed")[$display(frac(( − x^(2) ) ( − x^(2) )^(4), ( x^(2) )^(3)))$]


        
          + #math.equation(block: false, alt: "minus x to the power 4 y")[$− x^(4) y$]
+ #math.equation(block: false, alt: "x to the power 4")[$x^(4)$]


        
      
      
        + #math.equation(block: false, alt: "open parenthesis the fraction minus 2 x over y squared close parenthesis open parenthesis the fraction y squared over 3 x close parenthesis squared")[$( display(frac(− 2 x, y^(2))) ) attach(( display(frac(y^(2), 3 x)) ), t: 2)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction x squared z over 2 close parenthesis cubed open parenthesis the fraction minus 2 over x squared z close parenthesis cubed")[$attach(( display(frac(x^(2) z, 2)) ), t: 3) attach(( display(frac(− 2, x^(2) z)) ), t: 3)$]