Properties of Exponents

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Exercises 1 We are asked to simplify the expression shown below. (48⋅4-4)-2​ We will illustrate how to do this step by step, mentioning the properties of powers used. Note that there are many ways to start simplifying it. One way is to start by simplifying the expression inside the parentheses. Let's see how to do this. (48⋅4-4)-21. Use the Product of Powers Property, am⋅an=am+n(44)-22. Use the Power of a Power Property, (am)n=am⋅n4-83. Use the definition of a Negative Exponent, a-m=am1​481​4. Calculate power655361​
Exercises 2 Let's start by reviewing the Power of a Product Property.Let a and b be real numbers, and let m be an integer. To find the power of a product, find the power of each factor and multiply. (ab)m=ambm​ As we can see, this property can be used when we have a product being raised to a power. In those cases, we just need to raise each factor to that power and multiply. We can see an example below. (2⋅3)2=22⋅32​(2⋅3)2=4⋅9(2⋅3)2=36​
Exercises 3 Let's start by reviewing the Quotient of Powers Property.Let a be a nonzero real number, and let n and m be integers. To divide powers with the same base, subtract their exponents. anam​=am−n​ As we can see, this property can be used when powers with the same base are being divided. In those cases, we need to find the difference of the exponent of the numerator and denominator. The result is the common base raised to this difference. We can see an example below. 2628​=28−6​2628​=222628​=4​
Exercises 4 We are given the expressions shown below, and asked to simplify them and find the one that is different. We will label each of them in order to tell them apart. A) 33⋅36​B) 33+6​C) 36⋅3​D) 36⋅33​​ Before starting to simplify, notice that A and D are products with the same factors but in different order. According to the Commutative Property of Multiplication, this does not affect the product's value. Therefore, these expressions are equivalent. 33⋅36=36⋅33​ Now let's recall the Product of Powers Property. This property tells us that to multiply powers with the same base, we can simply add their exponents. Therefore, according to this property, the expressions A and B are equivalent. 33⋅36=33+6​ At this point, we can conclude that the expressions A, B, and C are all equivalent. To simplify them, we can simplify any of them. Let's simplify A) 33⋅36. 33⋅36am⋅an=am+n33+6Add terms39Calculate power19683 Hence, the three expression mentioned above simplify to 39=19683. Now, we just need to calculate C) 36⋅3, which, as we already know, is the different one. 36⋅3Multiply318Calculate power387420489 Now we can clearly see that this expression is not equivalent to the rest.
Exercises 5 To evaluate the given exponential expression, we will use the Zero Exponent Property. a0=1, for every nonzero number a​ Since -7​=0, we can simplify the given expression using the property above. (-7)0=1​
Exercises 6 To evaluate the given exponential expression, we will use the Zero Exponent Property. a0=1, for every nonzero number a​ Since 4​=0, we can evaluate the given expression using the property above. 40=1​
Exercises 7 To evaluate the given exponential expression, we will use the Negative Exponent Property. a-n=an1​, for every nonzero number a and integer n  ​ Since 5​=0 and -4 is an integer, we can evaluate the given expression using the property above. 5-4a-m=am1​541​Calculate power6251​
Exercises 8 To evaluate the given exponential expression, we will use the Negative Exponent Property. a-n=an1​, for every nonzero number a and integer n  ​ Since -2​=0 and -5 is an integer, we can evaluate the given expression using the property above. (-2)-5a-m=am1​(-2)51​Calculate power-321​Put minus sign in front of fraction-321​
Exercises 9 To evaluate the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to evaluate the given expression using these properties. 402-4​a0=112-4​1a​=a2-4a-m=am1​241​Calculate power161​
Exercises 10 To evaluate the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to evaluate the given expression using these properties. -905-1​a0=1-15-1​Put minus sign in front of fraction-15-1​1a​=a-5-1a-m=am1​-511​1a=1-51​
Exercises 11 To evaluate the given exponential expression, we will use the Negative Exponent Property. a-n=an1​, for every nonzero number a and integer n  ​ Let's try to evaluate the given expression using the property above. 6-2-3-3​Put minus sign in front of fraction-6-23-3​a-m=am1​-621​331​​Divide fractions-331​⋅162​Multiply fractions-3362​Calculate power-2736​ba​=b/9a/9​-34​
Exercises 12 To evaluate the given exponential expression, we will use the Negative Exponent Property. a-n=an1​, for every nonzero number a and integer n  ​ Let's try to evaluate the given expression using the property above. 3-4(-8)-2​a-m=am1​341​(-8)21​​Divide fractions(-8)21​⋅134​Multiply fractions(-8)234​(-a)2=a28234​Calculate power6481​
Exercises 13 To simplify the given exponential expression, we will use the Negative Exponent Property. a-n=an1​, for every nonzero number a and integer n  ​ Let's simplify the given expression using the property above. x-7=x71​​
Exercises 14 To simplify the given exponential expression, we will use the Zero Exponent Property. a0=1, for every nonzero number a​ Let's simplify the given expression using the property above. y0=1​
Exercises 15 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. 9x0y-3a0=19⋅1⋅y-3Identity Property of Multiplication9y-3a-m=am1​9y31​a⋅b1​=ba​y39​
Exercises 16 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. 15c-8d0a0=115c-8⋅1Identity Property of Multiplication15c-8a-m=am1​15c81​a⋅b1​=ba​c815​
Exercises 17 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. n02-2m-3​a0=112-2m-3​1a​=a2-2m-3a-m=am1​221​⋅m31​Calculate power41​⋅m31​Multiply fractions4m31​
Exercises 18 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. 32100r-11s​a0=1321⋅r-11s​Identity Property of Multiplication32r-11s​ca⋅b​=a⋅cb​r-11⋅32s​a-m=am1​r111​⋅32s​Multiply fractions32r11s​Calculate power9r11s​
Exercises 19 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. b-74-3a0​a0=1b-74-3⋅1​Identity Property of Multiplicationb-74-3​a-m=am1​b71​431​​Divide fractions431​⋅1b7​Multiply fractions43b7​Calculate power64b7​
Exercises 20 To simplify the given exponential expression, we will use the Negative Exponent Property. a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using the property above. 7-2q-9p-8​a-m=am1​721​⋅q91​p81​​Multiply fractions72q91​p81​​Divide fractionsp81​⋅172q9​Multiply fractionsp872q9​Calculate powerp849q9​
Exercises 21 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. 8-1z0x-722y-6​a0=18-1⋅1⋅x-722y-6​Identity Property of Multiplication8-1x-722y-6​ca⋅b​=a⋅cb​22⋅8-1x-7y-6​a-m=am1​22⋅811​⋅x71​y61​​Multiply fractions22⋅81x71​y61​​Divide fractions22⋅y61​⋅181x7​Multiply fractions22⋅y681x7​Calculate power4⋅y68x7​a⋅cb​=ca⋅b​y632x7​
Exercises 22 To simplify the given exponential expression, we will use the Zero and Negative Exponent Properties. Zero Exponent Property​a0=1, for every nonzero number aNegative Exponent Property​a-n=an1​, for every nonzero number a and integer n  ​ Let's try to simplify the given expression using these properties. 5-3z-1013x-5y0​a0=15-3z-1013x-5⋅1​Identity Property of Multiplication5-3z-1013x-5​ca⋅b​=a⋅cb​13⋅5-3z-10x-5​a-m=am1​13⋅531​⋅z101​x51​​Multiply fractions13⋅53z101​x51​​Divide fractions13⋅x51​⋅153z10​Multiply fractions13⋅x553z10​Calculate power13⋅x5125z10​a⋅cb​=ca⋅b​x51625z10​
Exercises 23 To simplify the given expression, we will use the Quotient of Power Property. anam​=am−n,​ for every nonzero number a and integers m and n​ Since 5​=0, we can use the property above to simplify the given expression. 5256​anam​=am−n56−2Subtract term54Calculate power625
Exercises 24 To simplify the given expression, we will use the Quotient of Power Property. anam​=am−n,​ for every nonzero number a and integers m and n​ Since -6​=0, we can use the property above to simplify the given expression. (-6)5(-6)8​anam​=am−n(-6)8−5Subtract term(-6)3(-a)3=-a3-63Calculate power-216
Exercises 25 To simplify the given expression, we will use the Product of Powers Property. am⋅an=am+n,​ for every nonzero number a and integers m and n​ Since -9​=0, we can use the property above to simplify the given expression. (-9)2⋅(-9)2am⋅an=am+n(-9)2+2Add terms(-9)4(-a)4=a494Calculate power6561
Exercises 26 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Product of Powers Property. 4-5⋅45am⋅an=am+n4-5+5Add terms40a0=11
Exercises 27 To simplify the given expression, we will use the Power of a Power Property. (am)n=amn, for every nonzero number a and integers m and n  ​ We can use the property above to simplify the given expression. (p6)4(am)n=am⋅np6(4)Multiplyp24
Exercises 28 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Power of a Power Property. (s-5)3(am)n=am⋅ns-5⋅3Multiplys-15a-m=am1​s151​
Exercises 29 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Product of Powers Property. 6-8⋅65am⋅an=am+n6-8+5Add terms6-3a-m=am1​631​Calculate power2161​
Exercises 30 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Product of Powers Property. -7⋅(-7)-4Write as a power(-7)1⋅(-7)-4am⋅an=am+n(-7)1+(-4)a+(-b)=a−b(-7)1−4Subtract term(-7)-3a-m=am1​(-7)31​(-a)3=-a3-731​Put minus sign in front of fraction-731​Calculate power-3431​
Exercises 31 To simplify the given expression, we will use the Properties of Exponents. For this exercise we will use the Quotient of Power Property. Then we will use the Product of Powers Property. Let's do it! x4x5​⋅xanam​=am−nx5−4⋅xSubtract termx1⋅xa1=ax⋅xa⋅a=a2x2
Exercises 32 To simplify the given expression, we will use the Properties of Exponents. For this exercise we will use the Product of Powers Property. Then we will use the Quotient of Power Property. Let's do it! z5z8⋅z2​am⋅an=am+nz5z8+2​Add termsz5z10​anam​=am−nz10−5Subtract termz5
Exercises 33 The microscope magnifies the 10-7 meter object 105 times. This means the magnified length of the object will be 105 times 10-7 meters. We can represent this as an expression and then simplify. 10-7⋅105am⋅an=am+n10-2a-m=am1​1021​Calculate power1001​ We have found that the magnified length is 1001​ meters.
Exercises 34 We are given the area and the width of a rectangle. We want to find its length. The formula for the area of a rectangle is length times width. A=ℓw​ Let's substitute the values we are given into this equation and then solve for the length. A=ℓwA=112a3b2, w=8ab112a3b2=ℓ(8ab) Solve for ℓ LHS/8ab=RHS/8ab8ab112a3b2​=ℓanam​=am−n8112​a2b=ℓSimplify quotient14a2b=ℓRearrange equation ℓ=14a2b We have found that the length of the rectangle is 14a2b.
Exercises 35 We will start by identifying the error. 24⋅25=(2⋅2)4+5 ×​ We can see here that the Product of Powers Property is incorrectly applied. When two exponential terms with the same base are multiplied, the base does not change in the answer. am⋅an=am+n​ The error occurred because in the answer the bases were multiplied together. Let's correct the error. 24⋅25am⋅an=am+n29 The correct solution is 29.
Exercises 36 To find where the error occurred, we will look at each step of the given solution. x4x5⋅x3​=x4x8​ ✓​ This first step is a correct application of the Product of Powers Property in the numerator. Let's look at the next step. x4x8​=x8/4 ×​ We have found the error. When exponential terms with the same base are divided, we can simplify by subtracting the exponents. This is the Quotient of Powers Property. anam​=am−n​ The error occurred because the exponents were divided instead of subtracted. Let's now fix this error and correctly simplify the expression. x4x5⋅x3​am⋅an=am+nx4x8​anam​=am−nx8−4Simplify powerx4 The correct simplification is x4.
Exercises 37 To simplify the given expression, we will use the Power of a Product Property. (-5z)3(ab)m=ambm(-5)3z3(-a)3=-a3-53z3Calculate power-125z3
Exercises 38 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Negative Exponent Property. Then we will use Power of a Product Property. Let's do it! (4x)-4a-m=am1​(4x)41​(ab)m=ambm44x41​Calculate power256x41​
Exercises 39 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Negative Exponent Property. Then we will distribute the exponent to the numerator and denominator, using the Power of a Quotient Property. Let's do it! (n6​)-2a-m=am1​(n6​)21​(ba​)m=bmam​n262​1​a/b1​=ab​62n2​Calculate power36n2​
Exercises 40 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by distributing the exponent to the numerator and denominator, using the Power of a Quotient Property. Let's do it! (3-t​)2(ba​)m=bmam​32(-t)2​(-a)2=a232t2​Calculate power9t2​
Exercises 41 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Negative Exponent Property. Then we will use Power of a Product Property. Let's do it! (3s8)-5a-m=am1​(3s8)51​(ab)m=ambm35(s8)51​(am)n=am⋅n35s401​Calculate power243s401​
Exercises 42 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Power of a Product Property. Then we will use Power of a Power Property. Let's do it! (-5p3)3(ab)m=ambm(-5)3(p3)3(am)n=am⋅n(-5)3p9(-a)3=-a3-53p9Calculate power-125p9
Exercises 43 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Negative Exponent Property. Then we will distribute the exponent to the numerator and denominator, using the Power of a Quotient Property. Let's do it! (-6w3​)-2a-m=am1​(-6w3​)21​(-a)2=a2(6w3​)21​bmam​=(ba​)m62(w3)2​1​(am)n=am⋅n62w6​1​a/b1​=ab​w662​Calculate powerw636​
Exercises 44 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by applying the Negative Exponent Property. Then we will use Power of a Product Property. Let's do it! (2r61​)-6a-m=am1​(2r61​)61​bmam​=(ba​)m(2r6)616​1​(ab)m=ambm26(r6)616​1​(am)n=am⋅n26r3616​1​Calculate power64r361​1​a/b1​=ab​164r36​1a​=a64r36
Exercises 45 We want to find the volume of a sphere with radius 2s2. V=34πr3​​ In the above equation, r represents radius. We can substitute our radius into the equation and simplify. V=34πr3​r=2s2V=34π(2s2)3​ Simplify right-hand side (a⋅b)m=am⋅bmV=34π(23⋅(s2)3)​(am)n=am⋅nV=34π(23⋅s6)​Calculate powerV=34π(8⋅s6)​Multiply V=332πs6​ We can see that this corresponds with option C. There is a chance some of the other expressions are correct as well, so we will check to see if they are equivalent to 332πs6​. Let's start with A. (24πs83s2​)-1 Simplify a-m=am1​3s224πs8​Calculate power3s216πs8​anam​=am−n 316πs6​ × We can see that option A does not equal 332πs6​, so it is incorrect. Let's try option B. (25πs6)(3-1) Simplify a-m=am1​325πs6​Calculate power 332πs6​ ✓ We can see that option B is also correct. Let's try option D. (2s)5⋅3πs​ Simplify (a⋅b)m=am⋅bm32⋅s5⋅3πs​a=1a​132⋅s5​⋅3πs​Multiply fractions332s5πs​a=a1332s5πs1​am⋅an=am+n 332πs6​✓ Option D is also correct. Let's try option E. (323πs6​)-1a-m=am1​3πs632​× This does not simplify to 332πs6​, so it incorrect. Finally, let's try option F. 332​πs5 Simplify a=1a​332​⋅1πs5​Multiply fractions 332πs5​× Option F is also incorrect. The only correct options are B, C, and D.
Exercises 46 We are given the equation t=2Dx2​ to describe the time it takes for molecules to diffuse. We are told that our x-value is 10-4 centimeters and our D-value is 10-5 square centimeters per second. We can substitute these into our equation and simplify. t=2Dx2​x=10-4, D=10-5t=2(10-5)(10-4)2​ Simplify right-hand side (am)n=am⋅nt=2(10-5)10-8​anam​=am−nt=210-13​Split into factorst=21​×10-13Write as a decimalt=0.5×10-13Write in scientific notation t=5×10-14 We have found that it will take 5×10-14 seconds for the ink to diffuse in water.
Exercises 47 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by distributing the exponent to the numerator and denominator. Let's do it! (3xy-42x-2y3​)4(ba​)m=bmam​(3xy-4)4(2x-2y3)4​(a⋅b)m=am⋅bm81x4(y-4)416(x-2)4(y3)4​(am)n=am⋅n81x4y-1616x-8y12​Write as a product of fractions8116​(x4x-8​)(y-16y12​) anam​=am−n anam​=am−n8116​x-8−4y12−(-16)a−(-b)=a+b8116​x-8−4y12+16Add and subtract terms 8116​x-12y28a-m=am1​8116​(x121​)y28b1​⋅a=ba​8116​(x12y28​)Multiply fractions81x1216y28​
Exercises 48 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by distributing the exponent to the numerator and denominator. Let's do it! (-2s-2t44s5t-7​)3(ba​)m=bmam​(-2s-2t4)3(4s5t-7)3​(ab)m=ambm(-2)3(s-2)3(t4)343(s5)3(t-7)3​(am)n=am⋅n(-2)3s-6t1243s15t-21​Write as a product of fractions(-2)343​(s-6s15​)(t12t-21​) anam​=am−n anam​=am−n(-2)343​s15−(-6)t-21−12a−(-b)=a+b(-2)343​s15+6t-21−12Add and subtract terms (-2)343​s21t-33a-m=am1​(-2)343​s21(t331​)a⋅b1​=ba​(-2)343​(t33s21​) Simplify (-a)3=-a3-2343​(t33s21​)Calculate power-864​(t33s21​)Put minus sign in front of fraction-864​(t33s21​)ba​=b/8a/8​ -8(t33s21​)a⋅cb​=ca⋅b​t33-8s21​
Exercises 49 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by distributing the exponent to the numerator and denominator. Let's do it! (4m-2n03m-5n2​)2⋅(9nmn4​)2(ba​)m=bmam​(4m-2n0)2(3m-5n2)2​⋅(9n)2(mn4)2​(ab)m=ambm16(m-2)2(n0)29(m-5)2(n2)2​⋅81n2m2(n4)2​(am)n=am⋅n16m-4n09m-10n4​⋅81n2m2n8​ Simplify Multiply fractions16m-4n081n29m-10n4m2n8​Commutative Property of Multiplication16⋅81m-4n0n29m-10m2n4n8​am⋅an=am+n16⋅81m-4n29m-8n12​ba​=b/9a/9​16⋅9m-4n2m-8n12​Multiply 144m-4n2m-8n12​Write as a product of fractions1441​(m-4m-8​)(n2n12​) anam​=am−n anam​=am−n1441​m-8−(-4)n12−2a−(-b)=a+b1441​m-8+4n12−2Add and subtract terms 1441​m-4n10a-m=am1​1441​(m41​)n10Multiply144m4n10​
Exercises 50 To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by distributing the exponent to the numerator and denominator. Let's do it! (x-23x3y0​)4⋅(5xy-8y2x-4​)3(ba​)m=bmam​(x-2)4(3x3y0)4​⋅(5xy-8)3(y2x-4)3​(ab)m=ambm(x-2)481(x3)4(y0)4​⋅125x3(y-8)3(y2)3(x-4)3​(am)n=am⋅nx-881x12y0​⋅125x3y-24y6x-12​ Simplify Multiply fractionsx-8125x3y-2481x12y0y6x-12​Commutative Property of Multiplication125x-8x3y-2481x12x-12y0y6​am⋅an=am+n 125x-5y-2481x0y6​Write as a product of fractions12581​(x-5x0​)(y-24y6​) anam​=am−n anam​=am−n12581​x0−(-5)y6−(-24)a−(-b)=a+b12581​x0+5y6+24Add terms 12581​x5y30ca​⋅b=ca⋅b​12581x5y30​
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