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not all roots are surds; little bit about ''e''
*(''x''<sup>-n</sup>) = 1 / ''x''<sup>n</sup>
*(''x''<sup>-n</sup>) = 1 / ''x''<sup>n</sup>
Fractional exponents give us [[surd]]s; for example, x<sup>0.5</sup> gives us the square root:
Fractional exponents 1/''n'' give us the n-th root; 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''
*(''x''<sup>0.5</sup>) · (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''
The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
The [[transcendental number]] ''[[e]]'' (2.7182818...) has the property that the [[derivative]] of the function ''e''<sup>''x''</sup> is ''e''<sup>''x''</sup>. This function is thus important in the solution of many types of differential equations.