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 $\displaystyle\,x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}$, where there are $\,n\,$ factors in the product. In particular, $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,$. That is, $\,x^{-1}\,$ is the reciprocal of $\,x\,$. |
When simplifying expressions involving exponent notation,
figure out the sign (plus or minus) of the expression first,
then figure out its size.
Recall that any even number ($2$, $4$, $6$, $\ldots$) of negative factors is positive.
Any odd number ($1$, $3$, $5$, $\ldots$) of negative factors is negative.
For example, consider
$\,(-2)^6\,$.
There are an even number ($6$) of negative factors, so the result is positive.
The size of the result is $\,2^6 = 64\,$.
Thus,
$\,(-2)^6 = 64\,$.
As a second example, consider
$\,(-2)^5\,$.
There are an odd number ($5$) of negative factors, so the result is negative.
The size of the result is $\,2^5 = 32\,$.
Thus,
$\,(-2)^5 = -32\,$.
Since exponents are done before multiplication,
$\,-2^4 = (-1)(2^4) = (-1)(16) = -16\,$.
Be careful!
The numbers
$\,-2^4\,$ and $\,(-2)^4\,$
represent different orders of operations,
and are different numbers!
The numbers
$\,-2^3\,$ and $\,(-2)^3\,$
represent different orders of operations,
but in this case they result in the same number!
If an expression is not defined, input “nd”.