SOLUTION: When working with exponents, is there any difference in operation if the exponents are whole numbers or fractions? In what sense do exponents and radicals behave exactly the same

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Question 279640: When working with exponents, is there any difference in operation if the exponents are whole numbers or fractions? In what sense do exponents and radicals behave exactly the same way?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
exponents and radicals act the same way as shown below:

16(1/4) = 2

root%284%2C16%29+=+2

the first equation is taking 16 and raising it to the (1/4) power.

the second equation is taking the 4th root of 16.

the 2 equations are equivalent.

x(1/n) is equivalent to root%28n%2Cx%29.

I can combine operations as follows:

I take the 4th root of 16 to get 2 and then I raise that to the 2d power I get 4.

this can be expressed as:

%28root%284%2C16%29%29%5E2+=+%282%29%5E2+=+4

this can also be expressed as:

2(4/2)

I can also do the following:

I take the square root of 2 and then raise that to the 4th power.

this can be expressed as:

%28root%282%2C2%29%29%5E4+=+4

this can also be expressed as:

2(4/2)

whether you take the root first and then raise that to the power, or you raise it to the power first and then take the root, you get the same answer.

2(4/2) is the same as root%282%2C2%5E4%29 is the same as %28root%282%2C2%29%29%5E4.