YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
|      | |     |
   \             /   
\    =   /

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

[ Up ] [ Compound Growth & Decay ] [ More Growth & Decay ] [ Logs and Exponents ] [ Logs & Exponentials - Summary ]

6. Exponents and Radicals - Alternate Version

This page derives the theory of radicals and exponents using properties of natural logarithms and the exponential functions.  Introducing the latter represent a complication, but it also simplifies the recognition of the domain of definition of exponents and radicals.

Natural logarithm and exponential functions with the aid of an electronic calculators

Calculator button exercises met above  suggest relationships between x²  and x³  buttons, the  a^x  or y^x  functions, and how to obtain a^x  = exp(x ln(a)) via a sequence of operations on the calculator. The latter requires a > 0 but allows x to be any real number. In the case of the natural e,  ln(e) = 1 and e^x  = exp(x ln(e)) = exp(x). So we can use the  e^x  button on the calculator to compute exp(x).  The latter function should not be confused with the EXP button present on some calculators.

 A. Natural Logarithms:

The following properties can be illustrated or confirmed  with an electronic calculator.

• ln(x) is defined only when x > 0 and not defined when x < 0.
  
• Fundamental Properties of Logarithms:
  
  ln(ab) = ln(a) + ln(b)
  
  when a > 0 and b > 0.
  
• ln(1) = 0 from  fundamental property or from a calculator.
• ln (1/a) = - ln(a)
  
• ln(aⁿ ) = n ln(a) for each whole number - demonstrate with numbers and even proofs for small values of n.
  
• ln(1/aⁿ ) = -n ln(a) for each whole number  n as ln(1/b) = ln

B. Exponential Function

The existence of the exponential function exp(x) can be assumed, and then computed with the aid of the e^x button on a calculator.

Alternatively, the points on the graph of y = ln(x) can be plotted and interpolated. Then exp(c) can be defined as the value of the  x-coordinate of the intersection of the horizontal line y = c with the graph y = ln(x).  The latter gives an example of the horizontal line method for calculating a function or defining an inverse function.  The use of this horizontal line method to calculate exp(c) and the use of the vertical line method to calculate ln(d) gives a pair of operations or functions, each of which reverses the other.  The formal definition of iverse function can be skipped or included here.  .

Main Properties

The following properties can be derived from the properties of the natural logarithm and the horizontal line method for defining exp(x) from the graph of y = ln(x).  Alternatively some or all properties can be assumed without proof, and the rest, if any, derived from the assumed ones.  In either case numerical examples are needed to empirically illustrate all the following properties.

• Domain: exp(x) = e^x is defined for all real numbers x, at least those we can enter into the display of an electronic calculator
• Fundamental Property of Exponentials:
  
   exp(a+b) = exp(a) * exp(b).
  
• exp(0) = 1
• exp(x) exp(-x) = 1 or  exp(-x) = 1 / exp(x)
• ln(exp(x)) = x when x is real - so the natural log function ln reverses the calculation of the exponential function exp (x),
• exp(ln(x)) = x when x > 0 - so exponential function exp  reverses the calculation of the  natural log function ln(x),
  

C. Powers with whole number exponents

D. Powers with real number exponents

Saying how to compute a number, defines it.

Suppose a is a positive number and x is a real number. Then compute the power  a^x  by the formula

 a^x  = exp(x ln(a))

In a^x  the number a > 0  is called the base and x is called the exponent.

Two properties a^x a^y   =  a^x+y  and  (a^x)^y   =  a^xy can be also suggested and verified in numerical exercises. Details of how these properties follow from the properties of the exponential function and natural logarithms follow.

E. m-th Roots (m whole) for positive real numbers

Suppose a is a positive number, and m is a whole number.

Assume a = 64 and m = 2  on first reading. Then assume  a = 64 and m = 3 on second reading.

We will try to solve the equation

  y^m = a

for y where y is a positive number.

Solution: The natural logarithm of both sides of the equation  y^m = amust be equal since   y^m and a are two different ways of writing or representing the same number on paper. Therefore

  ln(y^m ) = ln(a)

But the left side   ln(y^m ) = m ln (y) due to the fundamental property ln(pq) = ln(p) + ln (q) of natural logarithms when p and q are positive. Therefore

m ^.  ln(y) = ln(a)

The latter implies

and so

That is

should satisfy

  y^m = a

the m-th root of a is

F. Even Roots of Real Numbers

Saying how to compute a quantity defines it.

Recall a positive times a positive is positive, zero times zerio is zero and negative times a negative is positive. Therefore x ²   > 0 for all real numbers x.

Square Roots

Because x ²   > 0 for all real numbers x, the equation

x ² = a

only has solutions x when b > 0, that is only when a is non-negative.  Defining

as the nonnegative real solution of  x ² = a works only  if a is positive or zero. For a > 0, this solution is provided by the computation

 See above. The latter is called the principal or positive square root. The negative square root is obtained by taking the negative of the principal square root. It is given by

All Further Even Roots

Similarly, if n = 2m > 0 is an even whole number, then x ⁿ = x ^2m = x^m x ^m > 0 for all real numbers x. So   the equation

 x ^2m = a

only has a solution x when a > 0, that is only when a is non-negative. The solution x = 0 when a = 0. For a > 0, the positive solution, called the principal root, is

There is also a negative solution

_  nÖb

=

as the nonnegative real solution of  x ^2m = b works only  if b is positive.  This solution is then given by a^1/n = exp( (1/n)ln(b)). See above.

G. Odd Roots of Real Numbers

Saying how to compute a quantity defines it.

The identify x = sign(x) |x|.and consequences

Each real number x = sign(x) |x|.  For instance

• +5 = (+1) 5 as sign (5) = +1 and |+5| = 5 = distance of +5 = 5 to origin 0

• -4 = (-1) 4 as sign (4) = -1 and |-4| = 4 = distance of -4 to origin 0

• 0 = (0)(0) as sign(0) = 0 and |0| = 0 = distance of 0 to itself.

Now sign(x) = +1, 0 or -1. In all, three cases [sign(x)]² = 1. Therefore

x³ = [sign(x)]|x|³

In general,

x^2m+1 = [sign(x)]|x|^2m+1

since (+1)^2m+1 = +1 and (-1)^2m+1 = -1.

Cube Roots

For  a in  nonzero, the equation x³ = a implies or requires

|y|³ = |a|

Therefore

When a is positive,  taking y = |y| implies

y³ = |a| = a.

But when a is negative,  taking y = (-1)|y| gives

y³ =(-1)³ |y|³ = - |a| = a.

The foregoing is equivalent to saying

satisfies  y³ = a whenever a is nonzero. For a non-zero, that is a > 0 or a < 0, let

and let

All further Odd Roots:

 For  a in  nonzero, and n a whole number, the equation x²ⁿ⁺¹ = a implies or requires

|y|²ⁿ⁺¹ = |a|

Therefore

When a is positive,  taking y = |y| implies

y²ⁿ⁺¹ = |a| = a.

But when a is negative,  taking y = (-1)|y| gives

y³ =(-1)²ⁿ⁺¹ |y|³ = - |a| = a.

The foregoing is equivalent to saying

satisfies  y²ⁿ⁺¹ = a whenever a is nonzero. For a > 0 and a < 0

For a non-zero, that is a > 0 or a < 0, let

and let

Exercise: Sketch the graph of y = x³ for  -2 < x < 2.  The equation x³ = b  has one and only real solution real solution as the horizontal line y = b intersects the graph of y = x³ at most one point. According to the theory above, for each nonzero real number b let  b^1/3 and

is the real solution of  x ³ = b. Let

www.whyslopes.com
Lesson & Lesson Plans for
Sec IV (Maths 436)

a reference for learning and teaching functions, polynomials, solving linear systems, 
powers + exponents + bases + radicals (roots) , quadratic formulas, equations of straight lines

1A. Master Logic
1B. Problems Solving Method
2A Solve Linear Equations i
2B.Solve Linear Equation II
2C Use Equal Sign Properly
2D. Perfect Arithmetic Skills
3 Words & Symbols
3 Goals to Set for Students
4 Use Equations Backwardly
5. Master Functions & Relations
6. Exponents & Radicals I
6 Exponents & Radicals II
7. Straight Lines
8. Polynomials (x,/,+/-)
9. Quadratics
10 Prove it
13 Similarity Scale Factors
12 Trig & Triangles
14 Statistics
MEQ Intermediate Objectives
Remarks for Teachers

---

Sit down and study - no one else can do that for you.

Advice and Directions
What to do in School   & Why
How to Study Maths & Why

Preparing for Science 

Good News: If you can learn to follow a multi-step methods in any subject precisely, you should be able to do so in other subjects, as well. Hint: Start with arithmetic

Words Before Symbols: 
What is a Variable?
Level:  Secondary II to VI, or Grades 7 to 12)
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Complex number starter lesson  

Arithmetic Videos
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification

Arithmetic Videos

Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods

Fraction Starter Lesson
(simplify, multiply, divide & 
then add or subtract)

[Top of this Page][[Site Exit] [ Back ] [ Area Entrance ] [ Up ] [ Next ]
[Comments, Reactions, Feedback]
www.whyslopes.com

Favourite Sites:  BBC News  and  mathematics portion of  English National Curriculum  
Francais: ||Définition d'une variable || Algèbre || Arithmetique || Logique | | 

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby, a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.