question

6^3/2  can be evaluated two ways (both ways give the same answer):

 

Since the fraction is in the exponent, the numerator 3 means to cube the number, the denominator 2 means to find the square root.

 

If you cube first and then find the square root   --->   6³  =  216  and the square root of 216  =  14.7  (approximately).

 

If you find the square root first   --->   sqrt(6) = 2.4494...  and this number cubed is  14.7  (approximately).

 

So, if you have a fractional exponent, you can raise to a power first (the numerator) and then find the root (the denominator) of that answer, or you can find the root first and then raise that answer to the power.

\(6^{\frac{3}{2}}= \sqrt{6^3}=\sqrt{216}=6\sqrt6 \)

I seldom use approximations for square roots.

Because you can make it a number times a root, like what I have done above