Excellence in Algebra
Elementary Functions
Before we can be successful in science and calculus, we need some comfort
with algebra.
Here are two major algebra computations we do throughout this class (and
you will do throughout your career!)
Part 1, Functions
Lecture 1.0a, Excellence in Algebra: Exponents
exponential notation, and
Dr. Ken W. Smith
polynomial arithmetic
Sam Houston State University
Here we review exponential notation.
2013
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Elementary Functions
2013
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Smith (SHSU)
Elementary Functions
2013
Exponential Notation
More Exponential Notation
About three centuries ago, scientists developed abbreviations for
multiplication of a variable, replacing
x · x by x2 ,
x · x · x by x3 and
x · x · x · x · x by x5 , etc.
This is just an abbreviation! The exponent merely counts the number of
times the base appears in the product.
Our symbolic abbreviation involving exponents leads naturally to some
basic observations, sometimes called algebra “rules”.
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 the
exponents:
x·x·x
xxx
x3
=
=x
Similarly, 2 =
x·x
1xx
x
When we divide objects with the same base (x) we subtract the
exponents.
(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.
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Elementary Functions
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Multiplying by 1 leaves a number unchanged.
Since xn x0 = xn+0 = xn then multiplying by x0 leaves a number
unchanged.
So
x0 = 1
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(1)
We can also extend our exponent notation to rational exponents.
1
1
Since (x 2 )2 = x 2 ·2 = x1 = x then
√
1
x 2 = x.
More generally, denominators in exponents represent roots:
1
√
x q = q x.
Smith (SHSU)
Elementary Functions
(2)
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Practicing exponential notation
Kilobytes and powers of ten
Let’s practice this with two exercises
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 scientist
works 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:
2
Exercise. Simplify 8 3 .
Solution.
√
2
3
8 3 = (81/3 )2 = ( 8)2 = 22 = 4.
kilo- represent a thousand,
mega- represents a million and
3
2
Exercise. Simplify 4 .
Solution.
√
3
4 2 = (41/2 )3 = ( 4)3 = 23 = 8
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 .
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Elementary Functions
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Smith (SHSU)
Elementary Functions
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More bytes and exponents
More Exponential Notation
Exercise. How many digits are there in 2300 ?
To succeed in calculus, we need comfort with these algebra techniques.
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.
Practice these techniques throughout the semester, so that you can indeed
be comfortable with your algebra!
WolframAlpha gives
In the next slide presentation we practice problems involving exponents
and polynomials.
2300 = 2037035976334486086268445688409378161051468393665936250636140...
449354381299763336706183397376
(End)
You can check that this has 91 digits!
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Elementary Functions
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Smith (SHSU)
Elementary Functions
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Practicing with exponents
In this short lecture we practice some problems involving the algebra of
exponents and the algebra of polynomials.
Elementary Functions
Practice. (Some sample problems from an old quiz.)
√
3
Part 1, Functions
Lecture 1.0b, Excellence in Algebra: Practice Problems
x6
Simplify the expression √ .
x2
√
3
Solution. Rewrite 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.)
Dr. Ken W. Smith
Sam Houston State University
So
2013
√
3
√
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Elementary Functions
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More exponent practice
x2
=
x2
= x.
x
Elementary Functions
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More exponent practice
1
2x3 + x2
x4
Solution. Factor the numerator: 2x3 + x2 = (2x)x2 + (1)x2 = x2 (2x + 1).
x2
1
Also simplify 4 = 2 . So
x
x
2x3 + x2
x2 (2x + 1)
2x + 1
=
=
.
x4
x4
x2
(x6 ) 3 x−2
Simplify
x4
Solution. Simplify the numerator,
1
x2
(x6 ) 3 = x2 and x2 x−2 = 2 = 1.
x
So
1
(x6 ) 3 x−2
x2 x−2
1
=
= 4 or x−4
4
4
x
x
x
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Smith (SHSU)
x6
Elementary Functions
Simplify the expression
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Elementary Functions
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Review of polynomial arithmetic
More Exponential Notation
Once we learn the algebra abbreviation called “exponentiation”, we can
often use polynomial arithmetic (factoring polynomials) to simplify an
expression.
Let’s simplify the complex fraction
2x3
+
x4
1
x2
Dividing by
x2
Here are some other simplification problems involving factoring.
Recall that (A − B)(A + B) = A2 − B 2 and so we know how to factor a
difference of squares.
Simplify
1
x2
x2
1
is
the
same
as
multiplying
by
and
= 2 so
2
4
x
1
x
x
3
2
2x + x
2x3 + x2 x2
2x3 + x2
x4
=(
)(
)
=
1
x4
1
x2
2
x
x2 − 9
.
x+3
Solution. We recognize x2 − 9 = x2 − 32 as a difference of squares
(x − 3)(x + 3)
x2 − 9
=
= x − 3.
x+3
x+3
We factor the numerator and simplify.
x2 (2x + 1)
= 2x + 1.
Elementary
Functions
2
x
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Difference of squares
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Elementary Functions
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More algebra practice
One more....
x−9
Simplify √
x+3
Two solutions. (1) We could do this as a difference of squares!
√
(Think of x as the square of x.)
√
√
( x − 3)( x + 3) √
x−9
√
√
=
= x − 3.
x+3
x+3
x4 − 9
.
x2 + 3
Solution. We use the same pattern as before, recognizing that the
difference of squares A2 − B 2 = (A + B)(A − B).
Simplify
(2) Or maybe you were taught to multiply by the “conjugate” here:
√
x−9
x−9
x−3
√
= (√
)( √
)
x+3
x+3
x−3
(x2 − 3)(x2 + 3)
x4 − 9
=
= x2 − 3.
x2 + 3
x2 + 3
Simplify the denominator and then cancel:
√
√
x−9
x−3
(x − 9)( x − 3) √
(√
)( √
)=
= x − 3.
(x − 9)
x+3
x−3
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Elementary Functions
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Smith (SHSU)
Elementary Functions
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Comfort with algebra
To succeed in calculus, we need comfort with these algebra techniques.
Practice these techniques throughout the semester, so that you can indeed
be comfortable with your algebra!
(END)
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Elementary Functions
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