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Product Of The Powers

Theorem: \;\;\big(a^{x}\big)^y = a^{\displaystyle (xy)}\quad\forall\; x, y \in \mathbb{R} \text{ and }  a > 0

Prerequisites:
Exponentiation (definition)
Sum of the Powers Property (proof)
Continuity of exponential function

Proof:

Case I: When y\in\mathbb{Z}

\qquad\quad\begin{align}\big(a^{x}\big)^y &= a^x.a^x.a^x....a^x\qquad\;\;\text{($y$ times)}\\ &= a^{\displaystyle (x+x+x...x)}\qquad\text{(by sum of powers property)}\\ &= a^{\displaystyle (xy)}\end{align}

Case II: When y\in\mathbb{Q}

Let y=p/q, where p, q\in\mathbb{Z}

Let \quad a^{\tfrac{p}{q}x} = b\qquad\qquad\qquad\cdots\text{(1)}
\therefore\quad\;\; \big(a^{\tfrac{x}{q}}\big)^p = b\qquad\qquad\qquad\text{(from case I)}\\ \Rightarrow\quad\;\; a^{\tfrac{x}{q}} = b^{\tfrac{1}{p}}\qquad\qquad\qquad\;\text{(by definition of exponentiation)}

Raising both sides to the power q,

\therefore\quad\;\; \big(a^{\tfrac{x}{q}}\big)^q = \big(b^{\tfrac{1}{p}}\big)^q\\ \Rightarrow\quad\;\; a^{\tfrac{x}{q}q} = b^{\tfrac{q}{p}}\qquad\qquad\qquad\text{(from case I)}\\ \Rightarrow\quad\;\; a^{x} = b^{\tfrac{q}{p}}

Let q/p=r
\therefore\quad\quad a^{x} = b^r\\ \Rightarrow\quad\;\; \big(a^x\big)^{\tfrac{1}{r}} = b\qquad\qquad\qquad\text{(by definition of exponentiation)}\\ \Rightarrow\quad\;\; \big(a^x\big)^{\tfrac{1}{r}} = b = a^{\tfrac{p}{q}x}\qquad\quad\!\!\text{(from $(1)$)}

Case III: When y\in\mathbb{R}

If y is irrational, then in any neighbourhood of y, there exists infinitely many rational numbers. Since at these rational numbers, the given equality holds, hence by the continuity of the exponential functions, the equality must also hold at y.

Therefore the result holds at all real values of x and y.


Recommended:
Sum of the powers
Power of the product
Log of product
Log of powers

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