Download Link to ppt Lesson Notes - Mr Santowski`s Math Page

Math SL1 - Santowski
1
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product of powers: 34 x 36
34 x 36 = 34 + 6  add exponents if bases are equal
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quotient of powers: 39 ÷ 32
69 ÷ 62 = 69 - 2  subtract exponents if bases are equal
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power of a power: (32)4
(32)4 = 32 x 4  multiply powers
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power of a product: (3 x a)5
(3 x a)5 = 35 x a5 = 243a5  distribute the exponent
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power of a quotient: (a/3)5
(a/3)5 = a5 ÷ 35 = a5/243  distribute the exponent
2
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Evaluate 25 ÷ 25.
(i) 25 ÷ 25 = 25 – 5 = 20
OR
 (ii) 25 ÷ 25 = 32 ÷ 32 = 1
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Conclusion  20 = 1.
In general then b0 = 1
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Evaluate 23 ÷ 27.
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(i) 23 ÷ 27 = 23 – 7 = 2-4
(ii) 23 ÷ 27 = 8 ÷ 128 =
1/16 = 1/24
Thus  2-4 = 1/16 =
1/24
 In general then
 b-e = 1/be
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3
We will use the Law of Exponents to prove that 9½ =
√9.
 9½ x 9½ = 9(½ + ½) = 91
 Therefore, 9½ is the positive number which when
multiplied by itself gives 9 
 The only number with this property is 3, or √9 or 2 9
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So what does it mean? It means we are finding the
second root of 9  2 9
4
We can go through the same process to develop a
meaning to 271/3
 271/3 x 271/3 x 271/3 = 27(1/3 + 1/3 + 1/3) = 271
 Therefore, 271/3 is the positive number which when
multiplied by itself three times gives 27
 The only number with this property is 3, or 3 27
or the third root of 27
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1
n
In general b  n b
the nth root of b.
which means we are finding
5
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We can use our knowledge of Laws of Exponents to
help us solve bm/n
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ex. Rewrite 323/5 making use of the Power of powers
>>> (321/5)3
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so it means we are looking for the 5th root of 32
which is 2 and then we cube it which is 8
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In general, b  b
m
n
n
m


or b   b

m
n
1
n
m

 


 b
n
m
6
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The numbers 1,4,9,16,25,36,49,64,81,100,121,144
are important because ...
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Likewise, the numbers
1,8,27,64,125,216,343,512,729 are important
because ....
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As well, the numbers 1,16,81,256, 625 are important
because .....
7
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ex 1. Simplify the following expressions:
(i) (3a2b)(-2a3b2)
(ii) (2m3)4
(iii) (-4p3q2)3
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ex 2. Simplify (6x5y3/8y4)2
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ex 3. Simplify (-6x-2y)(-9x-5y-2) / (3x2y-4) and express answer
with positive exponents
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ex 4. Evaluate the following
(i) (3/4)-2
(ii) (-6)0 / (2-3)
(iii) (2-4 + 2-6) / (2-3)
8
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We will use the various laws of exponents to simplify
expressions.
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ex. 271/3
ex. (-320.4)
ex. 81-3/4
ex. Evaluate 491.5 + 64-1/4 - 27-2/3
ex. Evaluate 41/2 + (-8)-1/3 - 274/3
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ex. Evaluate (4/9)½ + (4/25)3/2
ex. Evaluate
3
4
8  16  125
4
3
9
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exponential functions have the general formula y = ax 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 = 2x
▪ x
y
-5.00000 0.03125
-4.00000 0.06250
-3.00000 0.12500
-2.00000 0.25000
-1.00000 0.50000
0.00000 1.00000
1.00000 2.00000
2.00000 4.00000
3.00000 8.00000
4.00000 16.00000
5.00000 32.00000
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(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
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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
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We will use a GDC (or WINPLOT) and investigate:
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(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 = 2x, y = 2x-3, and y
= 2x+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(2x) and y =
2x+3
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Go to this link from AnalyzeMath and work
through the tutorial on transformed
exponential functions
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Consider how y = ax changes  i.e. the
range, asymptotes, increasing/decreasing
nature of the function, shifting and reflecting
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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:
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(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
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(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
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(iii) using a quadratic formula when we can't factor quadratic
systems
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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:
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(iv) isolating a variable in trigonometric systems, by using an
"inverse" function ==> x = sin-1(a/b)
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so far, we haven't developed a strategy for solving
exponential equations, where the variable is present as an
exponent (ex. 3x = 1/27)
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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
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COMMON BASE
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so the equation 3x = 1/27 can be rewritten as 3x = 3-3
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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
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hence, x must equal -3, as both represent exponents
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From our graph, we can use the software to calculate
the intersection point of f(x) = 3x and f(x) = 1/27
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Thus we have our intersection at the point where x = -3,
which represents the solution to the equation 3x = 1/27
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Likewise, we can solve for other, more algebraically
difficult equations like 3x = 15
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Examples to work through in class: (work through
algebraically and verify graphically)
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ex 1. Solve 4x-8 = 27
ex 2. Solve 4x+2 = 512
ex 3. Solve 64x-2 = 164x
ex 4. Solve (3x²)/(3x) = 729
ex 5. Solve 3x+1 + 3x = 324
ex 6. Solve 42x - 8(4x) + 16 = 0
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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?
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From West Texas A&M - Integral Exponents
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From West Texas A&M - Rational Exponents
24
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HW
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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
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