Exponentials with positive integer exponents
The expression an means a⋅a⋅a⋅…⋅a⏟n times, where we
multiply a by itself n>0 times. Note that 1n=1 and 0n=0.
The three laws of exponents are:
- ab⋅ac=ab+c
- abac=ab−c, and
- (ab)c=abc.
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These laws are explained in the following video:
Viewed as a function of n, the function 2n grows very quickly after
a while, much faster than powers of n.
nn2n32n111224843927841664165251253210100100010242040080001,048,5763090027,0001,073,741,824…………
The same thing goes for any function an with a>1.
On the other hand, if a<1, then an shrinks rapidly:
(12)10=11024,(12)20=11,048,576,(12)30=11,073,741,824.
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