Math SL1 - Santowski 1

 product of powers: 3 4 x 3 6  3 4 x 3 6 =  add exponents if bases are equal  quotient of powers: 3 9 ÷ 3 2  6 9 ÷ 6 2 =  subtract exponents if bases are equal  power of a power: (3 2 ) 4  (3 2 ) 4 = 3 2 x 4  multiply powers  power of a product: (3 x a) 5  (3 x a) 5 = 3 5 x a 5 = 243a 5  distribute the exponent  power of a quotient: (a/3) 5  (a/3) 5 = a 5 ÷ 3 5 = a 5 /243  distribute the exponent 2

 Evaluate 2 5 ÷ 2 5.  (i) 2 5 ÷ 2 5 = 2 5 – 5 = 2 0 OR  (ii) 2 5 ÷ 2 5 = 32 ÷ 32 = 1  Conclusion  2 0 = 1.  In general then b 0 = 1  Evaluate 2 3 ÷ 2 7.  (i) 2 3 ÷ 2 7 = 2 3 – 7 = 2 -4  (ii) 2 3 ÷ 2 7 = 8 ÷ 128 = 1/16 = 1/2 4  Thus  2 -4 = 1/16 = 1/2 4  In general then  b -e = 1/b e 3

 We will use the Law of Exponents to prove that 9 ½ = √ 9.  9 ½ x 9 ½ = 9 (½ + ½) = 9 1  Therefore, 9 ½ is the positive number which when multiplied by itself gives 9   The only number with this property is 3, or √ 9 or  So what does it mean? It means we are finding the second root of 9  4

 We can go through the same process to develop a meaning to 27 1/3  27 1/3 x 27 1/3 x 27 1/3 = 27 (1/3 + 1/3 + 1/3) = 27 1  Therefore, 27 1/3 is the positive number which when multiplied by itself three times gives 27  The only number with this property is 3, or or the third root of 27  In generalwhich means we are finding the nth root of b. 5

 We can use our knowledge of Laws of Exponents to help us solve b m/n  ex. Rewrite 32 3/5 making use of the Power of powers >>> (32 1/5 ) 3  so it means we are looking for the 5th root of 32 which is 2 and then we cube it which is 8  In general, 6

 The numbers 1,4,9,16,25,36,49,64,81,100,121,144 are important because...  Likewise, the numbers 1,8,27,64,125,216,343,512,729 are important because....  As well, the numbers 1,16,81,256, 625 are important because

 ex 1. Simplify the following expressions:  (i) (3a 2 b)(-2a 3 b 2 )  (ii) (2m 3 ) 4  (iii) (-4p 3 q 2 ) 3  ex 2. Simplify (6x 5 y 3 /8y 4 ) 2  ex 3. Simplify (-6x -2 y)(-9x -5 y -2 ) / (3x 2 y -4 ) and express answer with positive exponents  ex 4. Evaluate the following  (i) (3/4) -2  (ii) (-6) 0 / (2 -3 )  (iii) ( ) / (2 -3 ) 8

 We will use the various laws of exponents to simplify expressions.  ex. 27 1/3  ex. ( )  ex /4  ex. Evaluate / /3  ex. Evaluate 4 1/2 + (-8) -1/ /3   ex. Evaluate  ex. Evaluate (4/9) ½ + (4/25) 3/2 9

 exponential functions have the general formula y = a x where the variable is now the exponent  so to graph exponential functions, once again, we can use a table of values and find points  ex. Graph y = 2 x ▪ x y           

 (i) no x-intercept and the y-intercept is 1  (ii) the x axis is an asymptote - horizontal asymptote at y = 0+  (iii) range { y > 0}  (iv) domain {xER}  (v) the function always increases  (vi) the function is always concave up  (vii) the function has no turning points, max or min points

 As seen in the previous slide, the graph maintains the same “shape” or characteristics when transformed  Depending on the transformations, the various key features (domain, range, intercepts, asymptotes) will change

 We will use a GDC (or WINPLOT) and investigate:  (i) compare and contrast the following: y = {5,3,2} x and y = {½, 1/3, 1/5} x  (ii) compare and contrast the following: y = 2 x, y = 2 x-3, and y = 2 x+3  (iii) compare and contrast the following: y = (1/3) x, and y = (1/3) x+3 and y = (1/3) x-3  (iv) compare and contrast the following: y = 8(2 x ) and y = 2 x+3

 Go to this link from AnalyzeMath and work through the tutorial on transformed exponential functionsfrom AnalyzeMath  Consider how y = a x changes  i.e. the range, asymptotes, increasing/decreasing nature of the function, shifting and reflecting

 solving means to find the value of the variable in an equation  so far we have used a variety of methods to solve for a variable:  (i) simply isolating a variable:  i.e in linear systems (i.e. 3x - 5 = x + 8) or  i.e in quadratic systems when the equation is 0 = a(x - h)² + k  (ii) factoring equations in quadratic systems (i.e. x² - 2x - 10 = 0 which becomes (x+3)(x-5)=0) and cubics, quartics, and even in rational functions  (iii) using a quadratic formula when we can't factor quadratic systems

 solving means to find the value of the variable in an equation  so far we have used a variety of methods to solve for a variable:  (iv) isolating a variable in trigonometric systems, by using an "inverse" function ==> x = sin -1 (a/b)  so far, we haven't developed a strategy for solving exponential equations, where the variable is present as an exponent (ex. 3 x = 1/27)

 we can adapt a simple strategy, which does allow us to isolate a variable, if we simply express both sides of an equation in terms of a common base  COMMON BASE  so the equation 3 x = 1/27 can be rewritten as 3 x = 3 -3  so, if the two sides of an equation are expressed with a common base, and both sides are equal in value, then it must follow that the exponents are equal  hence, x must equal -3, as both represent exponents

 From our graph, we can use the software to calculate the intersection point of f(x) = 3 x and f(x) = 1/27  Thus we have our intersection at the point where x = -3, which represents the solution to the equation 3 x = 1/27  Likewise, we can solve for other, more algebraically difficult equations like 3 x = 15

 Examples to work through in class: (work through algebraically and verify graphically)  ex 1. Solve 4 x-8 = 2 7  ex 2. Solve 4 x+2 = 512  ex 3. Solve 64 x-2 = 16 4x  ex 4. Solve (3 x² )/(3 x ) = 729  ex 5. Solve 3 x x = 324  ex 6. Solve 4 2x - 8(4 x ) + 16 = 0

 Ex 1. The value of an investment, A, after t years is given by the formula A(t) = 1280(1.085) t  (a) Determine the value of the investment in 5 and in 10 years  (b) How many years will it take the investment to triple in value?

 From West Texas A&M - Integral Exponents From West Texas A&M - Integral Exponents  From West Texas A&M - Rational Exponents From West Texas A&M - Rational Exponents 24

 HW  Ex 3A #1;  Ex 3B #1efhi;  Ex 3C #1fh, 2dg, 3cg, 4hip,  6dh, 7g, 8fh, 9dj, 10cjmnl, 11hklp, 12fip, 13  Ex 3D #1ag, 2d, 3ceg,4d, 5c, 6agj;  Ex 3E #1aef, 2ajk 25