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*''x''<sup>4</sup>, "''x'' to the fourth power," ''x'' · ''x'' · ''x'' · ''x''
*''x''<sup>4</sup>, "''x'' to the fourth power," ''x'' · ''x'' · ''x'' · ''x''
*''y''<sup>''n''</sup>, "''y'' to the n-th power," ''y'' · ''y'' · ''y'' · ... · ''y'', where ''y'' appears ''n'' times.
*''y''<sup>''n''</sup>, "''y'' to the n-th power," ''y'' · ''y'' · ''y'' · ... · ''y'', where ''y'' appears ''n'' times.
The superscripted value is called the ''exponent.''
The definition of exponentiation as repeated multiplication only makes sense when the exponent is a positive integer—what does it mean to say "x multiplied by itself half a time" or "minus three times?" However, mathematicians have found logical meanings for zero, negative, fractional, and even complex exponents. These meanings arise from the basic observation that
*(''x''<sup>a</sup>) · (''x''<sup>b</sup>) = ''x''<sup>(a + b)</sup>
We can show that the zeroth power of any nonzero number is 1
*(''x''<sup>0</sup>) · (''x''<sup>n</sup>) = ''x''<sup>0 + n</sup> = ''x''<sup>n</sup>
Dividing both sides by ''x''<sup>n</sup> we get
*(''x''<sup>0</sup) = 1
A negative exponent produces the [[reciprocal]] of the corresponding positive exponent:
*(''x''<sup>-n</sup>) · (''x''<sup>n</sup>) = ''x''<sup>n - n</sup> = ''x''<sup>0</sup> = 1
Dividing both sizes by (''x''<sup>n</sup>) we get
*(''x''<sup>-n</sup>) = 1 / ''x''<sup>n</sup>
Fractional exponents give us [[surd]]; for example, x<sup>0.5</sup> gives us the square root:
*(''x''<sup>0.5</sup>) · (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''