LAWS OF EXPONENTS - EXPLANATIONS AND PROOFS AND APPLICATION IN SOLVING PROBLEMS

Please study
Exponents before Laws of Exponents,
if you have not already done so.

There, we introduced the special notation of writing a (literal)
number being multiplied more than once, with examples.

We explained the terms :
Base, Exponent or Index, Exponential form, Expanded form.

We provided solved examples and problems for practice
with answers, to help to understand the conepts covered.

Here, we state the 7 Laws of indices and
the two Rules used for solving problems.

We also give the explanations and proofs of the 7 laws.

We provide a few Solved Examples and
Problems for Practice with Answers to help to apply
the 7 Laws and the 2 Rules in solving problems.

Laws of Exponents (or Indices)

Law 1 of Laws of Exponents :

Law 1: product of powers of the same base:
a^m x aⁿ = a^{m + n}

Law 2: power of a power:
(a^m)ⁿ = a^mn

Law 3: Powers with exponent being negative integer:
a^-n = 1⁄aⁿ

Law 4: quotient of powers of the same base:
a^m⁄aⁿ = a^{m - n}

Law 5: powers with exponent zero:
a⁰ = 1

Law 6: power of a product:
(ab)^m = a^m x b^m

Law 7: power of a quotient:
(a⁄b)^m = a^m⁄b^m

Rule 1 of Exponents :

Rule 1: In an Equation involving Exponents,
If the BASES (other than 0, 1, -1) of the L.H.S. and the R.H.S.
are equal, then, the EXPONENTS should be equal.
a ^m = aⁿ ⇒ m = n, where a ≠ 0, a ≠ 1 and a ≠ -1.

Rule 2 of Exponents :

Rule 2: In an Equation involving Exponents,
If the EXPONENTS (other than 0 ) of the L.H.S. and the R.H.S.
are equal, then, the BASES should be equal.
a ^m = b^m ⇒ a = b, where m ≠ 0.

You have to observe, understand, remember and apply
the above Laws in solving various problems.

The above Laws are valid for integral exponents (m and n).

You can apply the above Laws
(for integral exponents) by studying the
Sets of Solved Examples and doing Exercises,
given in the pages for which Links are given at the bottom of this page.

You can study the Explanations and Proofs of
the Laws given below in two Links
in the beginning (now) or in the middle or at the end of
applying the Laws to various problems
and getting thoroughly acquainted with them.

The best way to remember various Algebra Formulas
(or Math Formulas) is to apply them to a number of problems.
The best way to solve Algebra Problems (or Math Problems)
is to remember various formulas.

This is similar to the saying,
"if you like a thing, you will concentrate.
or if you concentrate on a thing, you will like it."

So, adopt the best way [i.e. trying to memorise the
formulas (Laws ) first or going to the problems first
or some way in between] that is suitable for you.

The above 7 Laws and the 2 Rules
are valid for Rational Number Exponents also.

The Link to the page which gives Explanation
of the Laws for Rational Exponents, and which
provides Links to Sets of solved examples and
exercises on their application to the
Rational Exponents, is given at the bottom.

Explanations and Proofs of the Laws :

For Explanations and Proofs of
the above Laws, go to

First Two Laws (Explanations and Proofs)

Explanations and Proofs of Remaining Laws

Set of Solved Examples on Exponents

For Solved Examples and Exercise
Problems on Application of the Laws,
Go to

Set1 of Examples and Exercise on Exponents

Set2 of Examples and Exercise on Exponents

Set3 of Examples and Exercise on Exponents

Set4 of Examples and Exercise on Exponents

Set5 of Examples and Exercise on Exponents

RATIONAL EXPONENTS :

The following Link takes you to the page
which gives Explanation of the Laws for
Rational Exponents, and provides Links
to Sets of solved examples and
exercises on their application to the
Rational Exponents.

Rational Exponents