Download - Elementary Functions Excellence in Algebra Here are two ...kws006/Precalculus/1.0_Algebra_Review_files/ 1_0...exponential notation, and polynomial arithmetic Here we review exponential

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.


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.


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)


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


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.


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!



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.

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http://www.wolframalpha.com

Elementary FunctionsPart 1, Functions

Lecture 1.0b, Excellence in Algebra: Practice Problems

Dr. Ken W. Smith

Sam Houston State University

2013


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.


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


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.


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.



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.


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.


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.


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)
