Properties of Exponents

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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.
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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.
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