Difference between revisions of "Exponentiation"
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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'' | ||
Revision as of 19:29, June 19, 2007
In mathematics, exponentiation is the name for the operation also called raising to a power. In simple cases, it refers to repeated multiplication. It is indicated by a superscript, a small number or expression written above the line:
- 26, "two to the sixth power," 2 · 2 · 2 · 2 · 2 · 2 = 64
- x4, "x to the fourth power," x · x · x · x
- yn, "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
- (xa) · (xb) = x(a + b)
We can show that the zeroth power of any nonzero number is 1
- (x0) · (xn) = x0 + n = xn
Dividing both sides by xn we get
- (x0</sup) = 1
A negative exponent produces the reciprocal of the corresponding positive exponent:
- (x-n) · (xn) = xn - n = x0 = 1
Dividing both sizes by (xn) we get
- (x-n) = 1 / xn
Fractional exponents give us surd; for example, x0.5 gives us the square root:
- (x0.5) · (x0.5) = x1 = x