DEFINITIONS: properties of exponents | |
base; exponent; power |
Let
$\,x\in\Bbb{R}\,$.
In the expression $\,x^n\,$, $\,x\,$ is called the base and $\,n\,$ is called the exponent or the power. |
positive integers |
If
$\,n\in\{1,2,3,\ldots\}\,$,
then $\,x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,$, where there are $\,n\,$ factors in the product. In this case, $\,x^n\,$ is just a shorthand for repeated multiplication. Note that $\,x^1 = x\,$ for all real numbers $\,x\,$. |
zero |
If
$\,x\ne 0\,$,
then
$\,x^0 = 1\,$.
The expression $\,0^0\,$ is not defined. |
negative integers |
If
$\,n\in\{1,2,3,\ldots\}\,$ and
$\,x\ne 0\,$,
then $\displaystyle\, \cssId{s26}{x^{-n}} \cssId{s27}{= \frac{1}{x^n}} \cssId{s28}{= \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}}$, where there are $\,n\,$ factors in the product. In particular, $\,\displaystyle \cssId{s31}{x^{-1} = \frac{1}{x}}\,$ for all nonzero real numbers $\,x\,$. That is, $\,x^{-1}\,$ is the reciprocal of $\,x\,$. |
With fractions, it looks like this:
$\displaystyle
\cssId{s36}{(\frac{a}{b})^{-1}}
\cssId{s37}{= \frac{1}{\frac{a}{b}}}
\cssId{s38}{= 1 \div \frac{a}{b}}
\cssId{s39}{= 1\cdot\frac{b}{a}}
\cssId{s40}{= \frac{b}{a}}$
That is, the reciprocal of $\displaystyle\,\frac{a}{b}\,$ is $\displaystyle\,\frac{b}{a}\,$.
Now that you've mired through this calculation once,
you'll never have to do it this long way again!
When a fraction is raised to the $\,-1\,$ power, it just flips.
The numerator becomes the new denominator, and the denominator becomes the new numerator.
As needed, input your answer as a diagonal fraction (e.g., 2/3), since you can't type horizontal fractions.