Properties of Exponents and Solving Exponential Equations Analytical, Numerical, and Graphical Connections Property of Exponents fr\1 r'\. 1. a m. a n = 2. a m a 11 ft\(\ Rewrite each expression below as a single base raised to a single exponent. 5 2x3. 5 s4x 2 x. 2 3x+4 2 2x+l. 2 x 5 5xr \( f).. ';)_ ).."t'?>x!'\ [!l_. Xt8 ] ".). (a 111 J 1 = C\, '. m t\.. ( y 2 23. 2 22x x x..... I '1J y:.. 4. (a)" (k) n, ' 5. a o = 'i. Ill ( "r: \ ft\ 6. a ;; = Q_, j 4 2x +2. 2 x1.. 2 s x+3 I'.,,... 5xt 3. :). ;)_?>t t3 ii >c.t3 S.x.t m 0
Simplify each expression below using the prope1iies of exponents. \ 1. (33 Y s j?> ",5.J rx., 1 ' '\ j,s,., 12a 3 ' l5a 5 ( Iba.. 5 y 1 C, '\ a., \5 5.. '/. (, T x 5 y 2 X y.j to 9 H, b y:. j. l5 6. U;:6r ) :j r "' s. [ :!Sb sa 9 0 1250.) ()., q () is 'b 2,. (. (l.. 2 1x '2. '1j '1 Rewrite each of the following expressions as a single exponential expression in the fom1 d 11
Rewrite each side of the following equations as a sin le ex onential ex ression of the ame base. TlJen, solve the equation by setting the exponents equal to each other. I. 27 x 2 _ 2x+5 (:/? _,._ :: l,.. yx t 5 :5. '!. X '= ;l_ I.\<.+ I() 3(o L\x.;: \ 0 = \b {x,::c l:,1 C. 42x7 = ( 2 xl } B:i 'l.. Jl.x lb. :::,3 3 '1 )( = 3x. '3 xt:l ::. 3.. 3 b. r)2;;4 3) 1 ( 7i} := (;. ')..)XA =. 3y'1 a:1'x 3x.1 J ix = 3x\ 2 5>< 8 Lx = uj 'x:, ';: "\ 1 The Graphical Connection of Solving Exponential Equations Consider the equatio 9 V Solve this equation by rewriting each side of the equation with the same base and then setting the exponents equal to each other. "'2> x.\ " ':: :l. * 1 = 'l:... \ Qx :: 4 = )
To solve exponential equations, both sides of the equation must be rewritten so that the bases are the H l "bl F 1 l x+3 3 x+4 1 Tl b are 2 and 3, neither of which can be rewritten as a power of the other. Therefore, at t1is point, the only way that we can solve this equation is to graph each side of the equation in the calculator and use the intersect function to find the point of intersection, which we showed previously is the solution to the equation. same. owever, sometunes t us 1s not poss1 e. or examp e, 111 t 1e equat10n = 1e ases 1. Enter 2 x + 3 into the YI in the calculator ' 2. Enter 3 x+4 1 into Y 2 in the calculator 3. Hit the GRAPH button so that both functions are'' now displayed. You must be able to CLEARLY see the point of intersection so there may need to be some adjustment made to the WINDOW that is being viewed. You'll see, I changed my window to be XMIN: 6 XMAX: 2 YMIN:,1 YMAX: 4 4. With the graphs displayed on the screen, go to the CALC menu by hitting the 2ND and TRACE keys. Choose option #5 intersect. 5. You will now be taken back to the home screen and "Mr. Blinky" appears. He is asking, "Am I on the first curve?" Hit ENTER to tell him "Yes." He then jumps to the second curve and asks, "Am I on the second curve?" Hit ENTER to tell him "Yes." Mr. Blinky then asks, "Do you now want me to guess what the intersection is?" Again, you hit ENTER to say, "Heck yeah, I want ou to QUess!!" 6. According to the calculator, then the solution to the equation<t::;:3!5[) Sometimes, there are two points of intersection of the graphs. You will have to find these one at a time. Mr. Blinky will find the point of intersection closest to where he is located. Use this method to find the solution of each of the equations below. 1. 2 x+l = 3 x+2 2. 2 x 3_2=2 x 2+2 N::: D.2>3Cf 'X.,:: 3,415 \ 'X 3. (1r2 +2=3 x 4 +4 & 0 8.Y, 15 '50
Exponents Practice Da Period. Rewrite each of the following expressions as a single base raised to a single power. Show your work. 1. 52 _ 5 3x 4 ( y 3 ixt )('+ 5 S ls 5xl 2. 42x3. ( 23 rx+4 l y2.)( ;. 5) :l,)l\ '4 " xt. l Q_ lo '"l (a + ':l 3. )(. 9 x5 (_ ) )(.; s )(.+ C. 'l,)c. \ 0 5 X\\ 1 4. 4 x5. 8 2x4 2 x+6 ()xs. ":3):1x.'t l J... )(+' 1'){\0 <.x1. l )C.t \.,._ 'i.. _,..,.. \ 1x :i. 'i \ ;t ')C....I,; 1. 5. 5 122.x. 25 x6 5,2.:l.)C.. ('5 :i.),c.'6 \ 5 15 t, m ':2.>C. _, 1... J 2 >x+> 6.125 4 X 5 ('5 ) 1x. 5. ) 1/_. '5, )(.. +:;>..; (1:> 1'1 w)''" \ '5, :i.,.1 :>..,.. Solve each of the following equations by first, rewriting each side of the equation as a single base raised to a single power. Then, set the exponents equal to each other and solving the equation for x. Remember, if this is not possible, you will need to solve the equation graphically on the calculator.?. 92x4 = 27 x3 (:} ) l '+ : (}:/')x! 2.>,',(o ': u o ix9 4 x '6 = 3 'X. '\ \)( 'J 8 2x+4 8. = 4 x+5 4 x3 (.l )l.)(.+ yt.?> (..)(I l. ::: ;l.,c.c. '\Y.t\'J_ l. 4X t' S = x.. t lo
For exercises 'I solve the exponential equations by rewriting each side of the equation as a power of the same base, if possible. If it is not possible to rew1ite each side as a power of the same base, solve the e uation using the graphing calculator. f. 54x+2 = 25 x8 54i.2..: ts 2.),c1 S4')( : 52.)(.\(. 1' )(.\ l; 2.><. _, I. h, (ff + 2.'X=IB V<= 9] 2 16 2 x 3 yc+1..= '34)a.)( J..l)( :: 14f)( )(C. ":: '8 4 ')( \)(; \ 4\ ID, 163x2 = B Sx.. )s>' :2.. = C?" )'» \2e ::, sx 48"!!>)( ')( 8/3 l't, 3x. 9 2x3 = 27 x+9 3 V. l:l) 1.v. 3: l)f"q x. 3x.: 31".1r7 B sx,. 3l>t\).; s ')( b :: """;). 7 ;: 3?> x:. x.