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

Please study
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

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.

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

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 = 7^{5 - 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.)
a^m⁄aⁿ = a^{n + (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:

a^m⁄aⁿ = a^{m - n}
where a is any non zero real number and m and n are positive integers.

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
a^m⁄a^m = a^{m - m}
⇒ 1 = a⁰ or a⁰ = 1.

This gives us Law 5 of Exponents.

powers with exponent zero:

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

Explanation and Proof of Law 6 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)
= a^m x b^m

This gives us Law 6 of Exponents.

power of a product:

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

Explanation and Proof of Law 7 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

(x⁄y)⁵ = (x⁄y) x (x⁄y) x (x⁄y) x (x⁄y) x (x⁄y)
= (x x x x x x x x x )⁄(y x y x y x y x y ) = x⁵⁄y⁵

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

This gives us Law 7 of Exponents.

power of a quotient :

(a⁄b)^m = a^m⁄b^m
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