Math SL1 - Santowski

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product of powers: 34 x 36 34 x 36 = 34 + 6 add exponents if bases are equal

quotient of powers: 39 ÷ 32

69 ÷ 62 = 69 - 2 subtract exponents if bases are equal

power of a power: (32)4

(32)4 = 32 x 4 multiply powers power of a product: (3 x a)5

(3 x a)5 = 35 x a5 = 243a5 distribute the exponent

power of a quotient: (a/3)5 (a/3)5 = a5 ÷ 35 = a5/243 distribute the exponent 

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Evaluate 25 ÷ 25.

(i) 25 ÷ 25 = 25 – 5 = 20 OR

(ii) 25 ÷ 25 = 32 ÷ 32 = 1

Conclusion 20 = 1.

In general then b0 = 1

Evaluate 23 ÷ 27.

(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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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

So what does it mean? It means we are finding the second root of 9

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2 9

2 9

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 or the third root of 27

In general which means we are finding the nth root of b.

5

3 27

nn bb 1

We can use our knowledge of Laws of Exponents to help us solve bm/n

ex. Rewrite 323/5 making use of the Power of powers >>> (321/5)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

mn

m

nn

m

n mn

m

bbbbb

1

or

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 .....

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ex 1. Simplify the following expressions: (i) (3a2b)(-2a3b2) (ii) (2m3)4

(iii) (-4p3q2)3

ex 2. Simplify (6x5y3/8y4)2

ex 3. Simplify (-6x-2y)(-9x-5y-2) / (3x2y-4) and express answer with positive exponents

ex 4. Evaluate the following (i) (3/4)-2

(ii) (-6)0 / (2-3) (iii) (2-4 + 2-6) / (2-3)

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We will use the various laws of exponents to simplify expressions.

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

  ex. Evaluate

 ex. Evaluate (4/9)½ + (4/25)3/2

9

34

43 125168

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

(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 = 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

Go to this link from AnalyzeMath and work through the tutorial on transformed exponential functions

Consider how y = ax 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. 3x = 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 3x = 1/27 can be rewritten as 3x = 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) = 3x and f(x) = 1/27

Thus we have our intersection at the point where x = -3, which represents the solution to the equation 3x = 1/27

Likewise, we can solve for other, more algebraically difficult equations like 3x = 15

Examples to work through in class: (work through algebraically and verify graphically)

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

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 - Rational Exponents

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HW

Ex 3A #1;Ex 3B #1efhi; Ex 3C #1fh, 2dg, 3cg, 4hip,6dh, 7g, 8fh, 9dj, 10cjmnl, 11hklp,

12fip, 13Ex 3D #1ag, 2d, 3ceg,4d, 5c, 6agj; Ex 3E #1aef, 2ajk

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