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###### Exercises
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Exercises 1 To simplify the given expression, we will use the Properties of Exponents. Let's do it! 32⋅34am⋅an=am+n32+4Add terms36
Exercises 2 To simplify the given expression, we will use the Properties of Exponents. Let's do it! (k4)-3(am)n=am⋅nk4(-3)a(-b)=-a⋅bk-12 Notice that we have been asked to write our answer using only positive exponents. Therefore, we will use the rule for the negative exponent. k-12↔k121​​
Exercises 3 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! (3s54r2​)3(ba​)m=bmam​(3s5)3(4r2)3​(a⋅b)m=am⋅bm33(s5)343(r2)3​(am)n=am⋅n33s1543r6​Calculate power27s1564r6​
Exercises 4 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! (4x-2y42x0​)2(ba​)m=bmam​(4x-2y4)2(2x0)2​(a⋅b)m=am⋅bm42(x-2)2(y4)222(x0)2​(am)n=am⋅n42x-4y822x0​Calculate power16x-4y84x0​a0=116x-4y84(1)​a⋅1=a16x-4y84​Calculate quotient4x-4y81​ Notice that we have been asked to write our answer using only positive exponents. Therefore, we will use the rule for the negative exponent. 4x-4y81​Commutative Property of Multiplication4y8x-41​ab-m=bma​x44y8​1​a/b1​=ab​4y8x4​
Exercises 5 A cube root is the number you can multiply by itself three times to get the value inside the radical. a⋅a⋅a=a3​ In this case, we need the number that can be multiplied three times to get 27. a=327​Split into factorsa=33⋅3⋅3​a⋅a⋅a=a3a=333​nan​=a{3}a=3 The cube root of 327​ is 3.
Exercises 6 We will split the base into perfect fourth-degree factors to simplify this fraction. Because the denominator of the exponent is 4, this will allow us to simplify the rational exponent. Let's start! (16)41​1​Write as a power(24)41​1​(am)n=am⋅n(2)4⋅41​1​4⋅4a​=a211​a1=a21​
Exercises 7 We will split the base into perfect cube factors to simplify this fraction. Because the denominator of the exponent is 3, this will allow us to simplify the rational exponent. Let's start! 51232​Split into factors((8)(8)(8))32​a⋅a⋅a=a3(83)32​(am)n=am⋅n83⋅32​a=3⋅3a​82Calculate power64
Exercises 8 When we are given a number in a radical, we are typically being asked for the principal root. When the index is even, this is the positive root. Since the index of 4​ is even, let's find its principal root. (4​)5Split into factors(2⋅2​)5a⋅a=a2(22​)5a2​=a25Calculate power32
Exercises 9 To graph the given exponential function, we will first make a table of values.x5xy=5x -15-10.2 0501 1515 25225 Let's now plot and connect the points (-1,0.2), (0,1), (1,5), and (2,25) with a smooth curve.We can see in the graph that the range is all real numbers greater than zero. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y>0​
Exercises 10 To graph the given exponential function, we will first make a table of values.x-2(61​)xy=-2(61​)x -1-2(61​)-1-12 0-2(61​)0-2 1-2(61​)1-0.333 2-2(61​)2-0.055 Let's now plot and connect the points (-1,-12), (0,-2), (1,-0.33), and (2,-0.055) with a smooth curve.We can see in the graph that the range is all real numbers less than zero. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y<0​
Exercises 11 To graph the given exponential function, we will first make a table of values.x6(2)x−4−1y=6(2)x−4−1 -16(2)-1−4−1=6(2)-5−1-0.813 06(2)0−4−1=6(2)-4−1-0.625 46(2)4−4−1=6(2)0−15 Let's now plot and connect the points (-1,-0.813), (0,-0.625), (4,5), and (5,11) with a smooth curve.We can see in the graph that the range is all real numbers greater than -1. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y>-1​
Exercises 12 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 1, y is multiplied by 3, which is greater than 1. This tells us that the table represents an exponential growth function.
Exercises 13 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 1, y is multiplied by 111​, which is less than 1. This tells us that the table represents an exponential decay function.
Exercises 14 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=3(1.88)tWrite as a sumy=3(1+0.88)t Our function matches the format of exponential growth. Here a=3 and r=0.88. y=3(1+0.88)t​ This means that the rate of growth, also known as the percent of change, is 88%.
Exercises 15 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=31​(1.26)tWrite as a sumy=31​(1+0.26)t Our function matches the format of exponential growth. Here a=31​ and r=0.26. y=31​(1+0.26)t​ This means that the rate of growth, also known as the percent of change, is 26%.
Exercises 16 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=80(53​)tba​=b⋅2a⋅2​y=80(106​)tWrite as a decimaly=80(0.6)tWrite as a differencey=80(1−0.4)t Our function matches the format of exponential decay. Here a=80 and r=0.4. y=80(1−0.4)t​ This means that the rate of decay, also known as the percent of change, is 40%.
Exercises 17
Exercises 18 Let's write the volume of the cedar chest. V=V=​Width⋅Length⋅Height664​  ⋅  1643​  ⋅  24351​​ Recall that na​ can be written as an1​. na​=an1​​ Notice that the denominator of the rational exponent is the index of the radical. Let's rewrite 664​. 664​=6461​​ Now we can use the properties exponents to simplify the expression. 664​⋅1643​⋅24351​Rewrite 664​ as 6461​6461​⋅1643​⋅24351​ Simplify Write as a power(26)61​⋅(24)43​⋅(35)51​(am)n=am⋅n26⋅61​⋅24⋅43​⋅35⋅51​b⋅ba​=a21⋅23⋅31am⋅an=am+n23+1⋅31Add terms24⋅31Calculate power16⋅3Multiply 48 The volume of the cedar chest is 48 ft3.
Exercises 19