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 comfort withalgebra.Here are two major algebra computations we do throughout this class (and youwill do throughout your career!)

exponential notation, and

polynomial arithmetic

Here we review exponential notation.

Smith (SHSU) Elementary Functions 2013 2 / 17

Excellence in Algebra

Before we can be successful in science and calculus, we need some comfort withalgebra.Here are two major algebra computations we do throughout this class (and youwill do throughout your career!)

exponential notation, and

polynomial arithmetic

Here we review exponential notation.

Smith (SHSU) Elementary Functions 2013 2 / 17

Excellence in Algebra

Before we can be successful in science and calculus, we need some comfort withalgebra.Here are two major algebra computations we do throughout this class (and youwill do throughout your career!)

exponential notation, and

polynomial arithmetic

Here we review exponential notation.

Smith (SHSU) Elementary Functions 2013 2 / 17

Excellence in Algebra

Before we can be successful in science and calculus, we need some comfort withalgebra.Here are two major algebra computations we do throughout this class (and youwill do throughout your career!)

exponential notation, and

polynomial arithmetic

Here we review exponential notation.

Smith (SHSU) Elementary Functions 2013 2 / 17

Excellence in Algebra

Before we can be successful in science and calculus, we need some comfort withalgebra.Here are two major algebra computations we do throughout this class (and youwill do throughout your career!)

exponential notation, and

polynomial arithmetic

Here we review exponential notation.

Smith (SHSU) Elementary Functions 2013 2 / 17

Excellence in Algebra

Before we can be successful in science and calculus, we need some comfort withalgebra.Here are two major algebra computations we do throughout this class (and youwill 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 for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

Exponential Notation

About three centuries ago, scientists developed abbreviations for multiplication ofa 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 thebase 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 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.

Smith (SHSU) Elementary Functions 2013 3 / 17

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

More Exponential Notation

Our symbolic abbreviation involving exponents leads naturally to some basicobservations, sometimes called algebra “rules”.

Multiplying by 1 leaves a number unchanged.Since xnx0 = xn+0 = xn then multiplying by x0 leaves a number unchanged.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

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

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

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

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

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

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

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

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

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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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

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 scientist works inbase 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 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 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 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 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 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 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 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 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 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 indeed becomfortable with your algebra!

In the next slide presentation we practice problems involving exponents andpolynomials.

(End)

Smith (SHSU) Elementary Functions 2013 8 / 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 indeed becomfortable with your algebra!

In the next slide presentation we practice problems involving exponents andpolynomials.

(End)

Smith (SHSU) Elementary Functions 2013 8 / 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 indeed becomfortable with your algebra!

In the next slide presentation we practice problems involving exponents andpolynomials.

(End)

Smith (SHSU) Elementary Functions 2013 8 / 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 indeed becomfortable with your algebra!

In the next slide presentation we practice problems involving exponents andpolynomials.

(End)

Smith (SHSU) Elementary Functions 2013 8 / 17