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 a^{n} by a^{-n}.

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

You can also see

a^{-1} = 1⁄a
(p⁄q)^{-n} = (q⁄p)^{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.
7^{5}⁄7^{2} = (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 7^{5}⁄7^{2} = 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}⁄a^{n} = a^{n + (m - n)}⁄a^{n}
= {a^{n} x a^{(m - n)} }⁄a^{n} (by applying Law 1)
= a^{(m - n)} (cancelling a^{n})

This gives us Law 4 of Exponents.

quotient of powers of the same base:

a^{m}⁄a^{n} = a^{m - 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 a^{m}⁄a^{m} = a^{m - m} ⇒ 1 = a^{0} or a^{0} = 1.

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) = 5^{4} x 6^{4};
(pq)^{3} = (pq) x (pq) x (pq)
= (p x p x p) x (q x q x q) = p^{3} x q^{3}.

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 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) = 7^{3}⁄9^{3}

(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 ) = x^{5}⁄y^{5}

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 Properties 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⁄a^{n}

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