Elementary FunctionsPart 1, Functions
Lecture 1.0a, Excellence in Algebra: Exponents
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 1 / 17
Excellence in Algebra
Before we can be successful in science and calculus, we need some comfortwith algebra.Here are two major algebra computations we do throughout this class (andyou will do throughout your career!)
exponential notation, and
polynomial arithmetic
Here we review exponential notation.
Smith (SHSU) Elementary Functions 2013 2 / 17
Exponential Notation
About three centuries ago, scientists developed abbreviations formultiplication of a variable, replacing
x · x by x2,x · x · x by x3 andx · x · x · x · x by x5, etc.
This is just an abbreviation! The exponent merely counts the number oftimes the base appears in the product.
This leads to some basic “rules” consistent with the abbreviation.
For example, x3 · x2 = (x · x · x) · (x · x) = x5
So if we multiply objects with the same base (x) we should add theexponents:
Similarly,x3
x2=
x · x · xx · x
=x
1
x
x
x
x= x
When we divide objects with the same base (x) we subtract theexponents.(x3)2 = (x · x · x)2 = (x · x · x)(x · x · x) = x · x · x · x · x · x = x6.Repeated exponentiation leads to multiplying exponents.
Smith (SHSU) Elementary Functions 2013 3 / 17
More Exponential Notation
Our symbolic abbreviation involving exponents leads naturally to somebasic observations, sometimes called algebra “rules”.
Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a numberunchanged.So
x0 = 1 (1)
We can also extend our exponent notation to rational exponents.Since (x
12 )2 = x
12·2 = x1 = x then
x12 =√x.
More generally, denominators in exponents represent roots:
x1q = q√x. (2)
Smith (SHSU) Elementary Functions 2013 4 / 17
Practicing exponential notation
Let’s practice this with two exercises
Exercise. Simplify 823 .
Solution.8
23 = (81/3)2 = (
3√8)2 = 22 = 4.
Exercise. Simplify 432 .
Solution.4
32 = (41/2)3 = (
√4)3 = 23 = 8
Smith (SHSU) Elementary Functions 2013 5 / 17
Kilobytes and powers of ten
Here is a problem common in computer science applications.We note that 210 = 1024 while 1000 = 103. So 210 ≈ 103
The electronics (on/off) of a computer means that a computer scientistworks in base two.Computer storage and computer memory is measured in powers of two.But the language of computer science uses traditional powers of ten:
kilo- represent a thousand,
mega- represents a million and
giga- a billion (etc.)
To a computer scientist, kilo- represents 210, not 103.Computer scientists approximate powers of 2 as powers of 10!Example. Approximate 230 as a power of ten: Since
230 = (210)3 and 210 ≈ 103
then 230 = (210)3 ≈ (103)3 = 109.
Smith (SHSU) Elementary Functions 2013 6 / 17
More bytes and exponents
Exercise. How many digits are there in 2300?
Solution. We write 2300 = (210)30 ≈ (103)30 = 1090.Now 1090 = 1× 1090 = 1 followed by 90 zeroes so it has 91 digits.Therefore 2300 should have 91 digits.
WolframAlpha gives
2300 = 2037035976334486086268445688409378161051468393665936250636140...
449354381299763336706183397376
You can check that this has 91 digits!
Smith (SHSU) Elementary Functions 2013 7 / 17
More Exponential Notation
To succeed in calculus, we need comfort with these algebra techniques.
Practice these techniques throughout the semester, so that you can indeedbe comfortable with your algebra!
In the next slide presentation we practice problems involving exponentsand polynomials.
(End)
Smith (SHSU) Elementary Functions 2013 8 / 17
Elementary FunctionsPart 1, Functions
Lecture 1.0b, Excellence in Algebra: Practice Problems
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 9 / 17
Practicing with exponents
In this short lecture we practice some problems involving the algebra ofexponents and the algebra of polynomials.
Practice. (Some sample problems from an old quiz.)
Simplify the expression3√x6√x2
.
Solution. Rewrite3√x6 as x2 since (this is the meaning of cube root!)
(x2)3 = x6.Rewrite
√x2 as x. (This is merely the meaning of square root.)
So3√x6√x2
=x2
x= x.
Smith (SHSU) Elementary Functions 2013 10 / 17
More exponent practice
Simplify(x6)
13 x−2
x4
Solution. Simplify the numerator,
(x6)13 = x2 and x2x−2 =
x2
x2= 1.
So
(x6)13 x−2
x4=
x2x−2
x4=
1
x4or x−4
Smith (SHSU) Elementary Functions 2013 11 / 17
More exponent practice
Simplify the expression2x3 + x2
x4
Solution. Factor the numerator: 2x3+x2 = (2x)x2+(1)x2 = x2(2x+1).
Also simplifyx2
x4=
1
x2. So
2x3 + x2
x4=
x2(2x+ 1)
x4=
2x+ 1
x2.
Smith (SHSU) Elementary Functions 2013 12 / 17
Review of polynomial arithmetic
Once we learn the algebra abbreviation called “exponentiation”, we canoften use polynomial arithmetic (factoring polynomials) to simplify anexpression.Let’s simplify the complex fraction
2x3 + x2
x41
x2
Dividing by1
x2is the same as multiplying by
x2
1and
x2
x4=
1
x2so
2x3 + x2
x41
x2
= (2x3 + x2
x4)(x2
1) =
2x3 + x2
x2
We factor the numerator and simplify.
x2(2x+ 1)
x2= 2x+ 1.
Smith (SHSU) Elementary Functions 2013 13 / 17
More Exponential Notation
Here are some other simplification problems involving factoring.
Recall that (A−B)(A+B) = A2 −B2 and so we know how to factor adifference of squares.
Simplifyx2 − 9
x+ 3.
Solution. We recognize x2 − 9 = x2 − 32 as a difference of squares
x2 − 9
x+ 3=
(x− 3)(x+ 3)
x+ 3= x− 3.
Smith (SHSU) Elementary Functions 2013 14 / 17
Difference of squares
Simplifyx4 − 9
x2 + 3.
Solution. We use the same pattern as before, recognizing that thedifference of squares A2 −B2 = (A+B)(A−B).
x4 − 9
x2 + 3=
(x2 − 3)(x2 + 3)
x2 + 3= x2 − 3.
Smith (SHSU) Elementary Functions 2013 15 / 17
More algebra practice
One more....
Simplifyx− 9√x+ 3
Two solutions. (1) We could do this as a difference of squares!(Think of x as the square of
√x.)
x− 9√x+ 3
=(√x− 3)(
√x+ 3)√
x+ 3=√x− 3.
(2) Or maybe you were taught to multiply by the “conjugate” here:
x− 9√x+ 3
= (x− 9√x+ 3
)(
√x− 3√x− 3
)
Simplify the denominator and then cancel:
(x− 9√x+ 3
)(
√x− 3√x− 3
) =(x− 9)(
√x− 3)
(x− 9)=√x− 3.
Smith (SHSU) Elementary Functions 2013 16 / 17
Comfort with algebra
To succeed in calculus, we need comfort with these algebra techniques.
Practice these techniques throughout the semester, so that you can indeedbe comfortable with your algebra!
(END)
Smith (SHSU) Elementary Functions 2013 17 / 17
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