EXPLANATIONS AND PROOFS
OF FIRST TWO LAWS WHICH
INVOLVE ADDING EXPONENTS

Please study
Laws of Exponents before Adding Exponents in Proofs
if you have not already done so.

Explanations and Proofs of Laws of Exponents

Explanation and Proof of Law 1 of Exponents with Adding Exponents

Did you answer the question (2) (iv) of Exercise in Exponents
as 5³ x 5² = (5 x 5 x 5) x (5 x 5) ?
You are right.
5³ means 5 is multiplied 3 times.
5² means 5 is multiplied 2 times.
5³ x 5² means 5 is multiplied (3 + 2) times.
That means 5³ x 5² = 5^{(3 + 2)}.

Similarly,
(3⁄5)⁴ x (3⁄5)⁵ = (3⁄5 x 3⁄5 x 3⁄5 x3⁄5) x (3⁄5 x 3⁄5 x 3⁄5 x 3⁄5 x 3⁄5)
= (3⁄5)⁹ = (3⁄5)^{(4 + 5)}

Generalising this, we get
a^m x aⁿ = (a x a x a x ...........m times) x (a x a x a x ...........n times)
= a x a x a x ...........(m + n) times = a^{(m + n)}
where a is any real number and m and n are positive integers.

This gives us Law 1 of Exponents.
with adding exponents.

product of powers of the same base:

a^m x aⁿ = a^{m + n}
where a is any real number and m and n are positive integers.

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Explanation and Proof of Law 2 of Exponents with Adding Exponents

How did you answer the question (2) (v)
of Exercise in 

Exponents ?

You might have written the first step as
(4⁵)³ = 4⁵ x 4⁵ x 4⁵.

Here, you might have expanded each factor of 4⁵
as 4 x 4 x 4 x ...5 times
and the final answer as 4 x 4 x 4 x ...15 times.

You are right.

But now we have Law 1,
which can be applied after the first step as follows.
(4⁵)³ = 4⁵ x 4⁵ x 4⁵ = 4^{5 + 5 + 5} = 4^{5 x 3} ( since 5 added 3 times is 5 x 3) = 4¹⁵.

Did you notice that (4⁵)³ = 4^{5 x 3} ?

Here Repeated Adding Exponents lead to Multiplying them.

Let us see one more example.

{(-3⁄4)³}⁴ = (-3⁄4)³ x (-3⁄4)³ x (-3⁄4)³ x (-3⁄4)³
= (-3⁄4)^{3 + 3 + 3 + 3} ( By Applying Law 1, above)
= (-3⁄4)^{3 x 4} (since 3 added 4 times is 3 x 4)

Again, did you observe that {(-3⁄4)³}⁴ = (-3⁄4)^{3 x 4} ?

Again, Repeated Adding Exponents lead to Multiplying them.

Generalising this, we get
(a^m)ⁿ = a^m x a^m x a^m x .......n times
= a^{m + m + m + ......n terms} (By Applying Law 1, above)
= a^{m x n} (since m added n times is m x n) = a^mn
where a is any real number and m and n are positive integers.

This gives us Law 2 of Exponents.
with adding exponents.

power of a power:

(a^m)ⁿ = a^mn
where a is any real number and m and n are positive integers.

You may observe:
(a^m)ⁿ = a^mn = a^nm ( since, mn = nmi.e. commutative law) = (aⁿ)^m.

thus, corollary of Law 2:

(a^m)ⁿ = (aⁿ)^m.

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