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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

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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.

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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.

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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.

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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

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