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Negative exponents can be a bit confusing at first, but they are just a different way of expressing fractions. For example, if you have a number like 2 with a negative exponent, such as \(2^{-3}\), it means you take the reciprocal of the base raised to the positive of that exponent.
In this case, \[2^{-3}\ = \frac{1}{2^3} = \frac{1}{8}.\] This rule generally applies to any negative exponent.
For another example, let’s look at something like \16^{-\frac{1}{4}}\ from the problem: \[16^{-\frac{1}{4}} = \frac{1}{16^{\frac{1}{4}}}.\]
So whenever you encounter negative exponents, just remember you are dealing with reciprocals!

Fractional exponents, sometimes called rational exponents, are exponents that are fractions. They are another way to represent roots.
For instance, \16^{\frac{1}{4}}\ is the same as saying 'the fourth root of 16'.
Let's break it down: \16^{\frac{1}{4}}\ means you find a number which, when raised to the power of 4, gives you 16. We know that \2^4 = 16\.
So, \[16^{\frac{1}{4}} = 2.\] This can be generalized for any number and fractional exponent. For instance, \x^{\frac{m}{n}}\ means the n-th root of \x\ raised to the m-th power.

Imaginary numbers come into play when we need to take the root of a negative number.
In the problem, we see \((-16)^{\frac{1}{4}}\), which involves the fourth root of a negative number. In real numbers, this cannot be solved since a negative number cannot be a result of an even root. This is where imaginary numbers step in.
Imaginary numbers are built around the unit imaginary number, denoted by \(i\), where \[i = \sqrt{-1}.\]
So, when you deal with an expression like \((-16)^{\frac{1}{4}}\), it translates to finding the fourth root of \-1\, and then multiplying it by the fourth root of 16.
This gives: \[ (-16)^{\frac{1}{4}} = ((-1) \times 16)^{\frac{1}{4}} = (16)^{\frac{1}{4}} \times (-1)^{\frac{1}{4}} = 2 \times \frac{\rm 1 + i\rm \theta}{2}\] Here, keep in mind that \i\ satisfies \i^2 = -1 and apply the root properties accordingly.