# EXPLANATIONS AND PROOFS OF PROPERTIES OF EXPONENTS WITH EXAMPLES OF NUMERICALS AND LITERALS

Properties of Exponents
before their proofs.
Proofs of The first two Laws
were covered in
Proofs of the First Two Laws.
The proofs of other (Laws)
Properties of Exponents are given here.

## Explanations and Proofs of (Laws) Properties of Exponents

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

and remember large chunks of information
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
giving up your fun, such as TV, surfing the net,
playing video games or going out with friends!

Know more about the Speed Study System.

### Explanation and Proof of Law 3 of Properties of Exponents

Powers with exponent being negative integer:
We denote the multiplicative inverse of an by a-n.

a-n = 1⁄an
where a is any real number (≠ 0) and n is a positive integer.

You can also see

a-1 = 1⁄a
(pq)-n = (qp)n.
where p and q are integers (≠ 0).

are special cases of the same Law.

### Explanation and Proof of Law 4 of Properties of Exponents

Look at the question (2) (vi) of Exercise in Exponents.
75⁄72 = (7 x 7 x 7 x7 x7)⁄(7 x 7) = 7 x 7 x 7.(by cancelling 7 x 7).
Congratulations for soving it correctly.

Here, the point is to observe 75⁄72 = 75 - 2

More generally:
We have m = m + n - n (Here is n is added and subtracted to m)
= n + (m - n) (Here m and -n are written at one place.)
aman = an + (m - n)an
= {an x a(m - n) }⁄an (by applying Law 1) = a(m - n) (cancelling an)

This gives us Law 4 of Exponents.

quotient of powers of the same base:

aman = am - n
where a is any non zero real number and m and n are positive integers.

## Research-based personalized Math Help tutoring program : Properties of Exponents

Here is a resource for Solid Foundation in
Math Fundamentals from Middle thru High School.
You can check your self by the

### FREE TRIAL.

Are you spending lot of money for math tutors to your
child and still not satisfied with his/her grades ?

Do you feel that more time from the tutor and
more personalized Math Help to identify and fix
the problems faced by your child will help ?

Here is a fool proof solution I strongly recommend
and that too With a minuscule fraction of the amount
you spent on tutors with unconditional 100% money
back Guarantee, if you are not satisfied.

### SUBSCRIBE, TEST, IF NOT SATISFIED, RETURN FOR FULL REFUND

It is like having an unlimited time from an excellent Tutor.

It is an Internet-based math tutoring software program
that identifies exactly where your child needs help and
then creates a personal instruction plan tailored to your
child’s specific needs.

If your child can use a computer and access
the Internet, he or she can use the program.
And your child can access the program anytime
from any computer with Internet access.

### Unique program to help improve math skills quickly and painlessly.

There is an exclusive, Parent Information Page provides YOU
with detailed reports of your child’s progress so you can
monitor your child’s success and give them encouragement.
These Reports include

• Time spent using the program
• Assessment results
• Personalized remediation curriculum designed for your child
• Details the areas of weakness where your child needs additional help
• Provides the REASONS WHY your child missed a concept
• List of modules accessed and amount of time spent in each module
• Quiz results
• Creates reports that can be printed and used to discuss issues with your child’s teachers
These reports are created and stored in a secure section
of the program, available exclusively to you, the parent.
The section is accessed by a password that YOU create and use.
No unauthorized users can access this information.

### Personalized remediation curriculum designed for your child

Thus The features of this excellent Tutoring program are

• Using detailed testing techniques
• Identifing exactly where a student needs help.
• Its unique, smart system pinpointing precise problem areas -
• slowly and methodically guiding the student
• raising to the necessary levels to fix the problem.

### Not a “one-size-fits-all” approach!

Its research-based results have proven that
it really works for all students! in improving
math skills and a TWO LETTER GRADE INCREASE in
math test scores!,if they invest time in using
the program.

Proven for More than 10,000 U.S. public school
students who increased their math scores.

### Explanation and Proof of Law 5 of Properties of Exponents

Consider a special case of Law 4 when m = n.
Replacing n with m in L.H..S. and R.H.S., of Law 4, we get
amam = am - m
⇒ 1 = a0 or a0 = 1.

This gives us Law 5 of Properties of Exponents.

powers with exponent zero:

a0 = 1
where a is any non zero real number.

### Explanation and Proof of Law 6 of Properties of Exponents

Consider the following examples:
(5 x 6)4 = (5 x6) x (5 x6) x (5 x6) x (5 x6)
= (5 x 5 x 5 x 5) x (6 x 6 x 6 x 6) = 54 x 64;
(pq)3 = (pq) x (pq) x (pq)
= (p x p x p) x (q x q x q) = p3 x q3.

similarly, (ab)m = (ab) x (ab) x (ab) x .....m times
= (a x a x a x .....m times) x (b x b x b x .....m times)
= am x bm

This gives us Law 6 of Exponents.

power of a product:

(ab)m = am x bm
where a and b are real numbers and m is a positive integer.

### Explanation and Proof of Law 7 of Properties of Exponents

See the following examples:
(7⁄9)3 = (7⁄9) x (7⁄9) x (7⁄9) = (7 x 7 x 7)⁄(9 x 9 x 9) = 73⁄93

(xy)5 = (xy) x (xy) x (xy) x (xy) x (xy)
= (x x x x x x x x x )⁄(y x y x y x y x y ) = x5y5

in general, (ab)m = (ab) x (ab) x (ab) x ......m times
= (a x a x a x....m times)⁄(b x b x b x.....m times)
= ambm

This gives us Law 7 of Properties of Exponents.

power of a quotient :

(ab)m = ambm
where a and b are non zero real numbers and m is a positive integer.

NOTE:All the Laws defined for positive integers, can be extended
to Negative integers also with the idea a-n = 1⁄an

## Progressive Learning of Math : Properties of Exponents

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.