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Section 6.4 Closing Ideas

There is a balance between writing out all the steps and just using formulas. Just as with other algebraic steps (such as combining like terms), eventually the expectation is that both the person writing things out and the person reading it will be comfortable enough that all of the individual steps will not need to be written out.

But even as you get more experience, there will come moments when you can't quite remember the formula. When this happens, there is a mantra you can use to help jog your memory: "When in doubt, write it out!" What this means is that you can always fall back on the basic ideas to recover these formulas.

Remember that exponents, in their most basic form, are just a shorthand notation for repeated multiplication. After that, it's just logic.

The Product Rule for Exponents:

\begin{equation*} x^m \cdot x^n = \underbrace{\underbrace{(x \cdot x \cdots x)}_{\text{$m$ times}} \cdot \underbrace{(x \cdot x \cdots x)}_{\text{$n$ times}}}_{\text{$m +n$ times}} = x^{m+n} \end{equation*}

The Power Rule for Exponents:

\begin{equation*} \left( x^m \right)^n = \underbrace{ \underbrace{(x \cdot x \cdots x)}_{\text{m times}} \cdot \underbrace{(x \cdot x \cdots x )}_{\text{$m$ times}} \cdots \underbrace{(x \cdot x \cdots x )}_{\text{$m$ times}}}_{\text{$n$ groups of $m$ times}} = x^{mn} \end{equation*}