what are the similarities between radicals and rational exponent notation?

Answer 1

the rational exponents are a way to write radicals:
a^{1/2} = sqrt (a)
a^{1/3} = cube root of a

a^{2/3}= ( a^{1/3} )^2 = (cube root of a)^2 .

Answer 2

Rationa exponents are just another way of writing radicals or surds.
N^(1/m) = the mth root of N
for example, 5^(1/2) = sqrt 5
8^(1/3)= cube root of 8
etc
It also works for expressions such as N^(a/b) which is the same as (N^a)^(1/b) or (N^(1/b))^a
For example 8^(2/3) = (cube root of 8)^2 = 2^2 = 4
Radical exponent notation is more recent than surd notation, came into common use with Napier's work in the sixteenth century. Surd notation is much older, it can be traced back to the time of Pythatgoras. Exponent notation is a lot more flexible and powerful, also allows for non-rational exponents as in logs. If you plot the curve y = x^n for any constant n, you'll see that the curve is continuous, so it is defined for all values of n, even non rational values. Thus we can have expressions such as 2^pi or even e^(pi)
Just a minute and I'll get you some references where you can find out more.
http://en.wikipedia.org/wiki/Numerical
http://en.wikipedia.org/wiki/Exponent
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e.html
http://www.homeschoolmath.net/teaching/irrational_numbers.php