the parentheses/brackets force the operations to be performed in a particular sequence. Thus,

[(-4)^2]^3/4 = [16]^3/4 = 8

is the only correct solution.


The parentheses don't force the operations to be done in that particular order. You could do what Emily did and multiply the exponents

Young Emily multiplied two times 3/4, and got 3/2. She raised -4 to the power three, getting -64. She then took its square root, getting -8i.


The problem she made was that she multiplied them and then simplified. If she multiplied two times 3/4 and got 6/4 and then raised -4 to the sixth power getting 4096. Taking the fourth root of 4096 she would have gotten the correct answer of 8. Most of the time it doesn't do any harm to simplify a fraction, however when the fraction is in the exponent, you need to be real careful before you simplify. This is an example of a case where you should not simplify.