DEFINITIONS: properties of exponents | |
base; exponent; power |
Let
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$\,x\in\Bbb{R}\,$. In the expression [beautiful math coming... please be patient] $\,x^n\,$, $\,x\,$ is called the base and $\,n\,$ is called the exponent or the power. |
positive integers |
If
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$\,n\in\{1,2,3,\ldots\}\,$, then [beautiful math coming... please be patient] $\,x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,$, where there are $\,n\,$ factors in the product. In this case, [beautiful math coming... please be patient] $\,x^n\,$ is just a shorthand for repeated multiplication. Note that [beautiful math coming... please be patient] $\,x^1 = x\,$ for all real numbers $\,x\,$. |
zero | If
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$\,x\ne 0\,$, then
$\,x^0 = 1\,$. The expression $\,0^0\,$ is not defined. |
negative integers |
If
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$\,n\in\{1,2,3,\ldots\}\,$ and
$\,x\ne 0\,$,
then [beautiful math coming... please be patient] $\displaystyle\,x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}$, where there are $\,n\,$ factors in the product. In particular, [beautiful math coming... please be patient] $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,$. That is, [beautiful math coming... please be patient] $\,x^{-1}\,$ is the reciprocal of $\,x\,$. |
With fractions, it looks like this:
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$\displaystyle
(\frac{a}{b})^{-1} = \frac{1}{\frac{a}{b}} = 1 \div \frac{a}{b} = 1\cdot\frac{b}{a} = \frac{b}{a}$
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.