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Section A.12 Powers

In the following, \(x\) and \(y\) are arbitrary real numbers, and \(q\) is an arbitrary constant that is strictly bigger than zero.

  • \(q^0=1\)
  • \(q^{x+y}=q^xq^y\text{,}\) \(q^{x-y}=\frac{q^x}{q^y}\)
  • \(q^{-x}=\frac{1}{q^x}\)
  • \(\big(q^x\big)^y=q^{xy}\)
  • \(\lim\limits_{x\rightarrow\infty}q^x=\infty\text{,}\) \(\lim\limits_{x\rightarrow-\infty}q^x=0\) if \(q \gt 1\)
  • \(\lim\limits_{x\rightarrow\infty}q^x=0\text{,}\) \(\lim\limits_{x\rightarrow-\infty}q^x=\infty\) if \(0 \lt q \lt 1\)
  • The graph of \(2^x\) is given below. The graph of \(q^x\text{,}\) for any \(q \gt 1\text{,}\) is similar.