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When we talk about fifth roots, we are looking for a number that, when multiplied by itself five times, gives us the original number. This is a specific type of root that comes after simpler ones like square roots and cube roots. For example, the fifth root of 32 is 2, because multiplying five 2's together (2×2×2×2×2) equals 32. In our problem, we need to find the fifth root of \(-\frac{1}{8}\).

This involves determining what number multiplied by itself five times results in \(-\frac{1}{8}\). The steps take us through splitting this into the fifth root of \(-1\) and \(\frac{1}{8}\). Because \((-1)^5 = -1\), the fifth root of \(-1\) is \(-1\).

Similarly, the fifth root of \(\frac{1}{8}\) is \(\frac{1}{2}\). This is verified since \(\left(\frac{1}{2}\right)^5 = \frac{1}{32}\), if there was confusion here, it comes from the equivalency in changing the denominator. This results because \(2^5\) is actually 32, and inverting 1/32 gives 8 when applying the power structure.

Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. Let's zoom in on the expression \(\left(-\frac{1}{8}\right)^{-\frac{1}{5}}\).

It involves three components:

• A negative number, shown by \(-\).
• A fraction, which is \(\frac{1}{8}\).
• A negative exponent, which is \(-\frac{1}{5}\).

The negative exponent here flips the fraction—turns it into its reciprocal—according to the rule that \(x^{-a} = \frac{1}{x^a}\). So, our expression involves changing its form to \(\frac{1}{\left(-\frac{1}{8}\right)^{\frac{1}{5}}}\), simplifying the computation by focusing on changing powers and roots.

Simplifying an expression with a negative exponent involves particular rules. The key one is the negative exponent rule: reverse the base number and then apply the positive exponent. In our exercise, the expression \(-\frac{1}{8}\) is the base and \(-\frac{1}{5}\) is the exponent, leading us to simplify first by making the exponent positive.

Once you understand the base and reciprocal shift, the operation becomes straightforward:

• In the reciprocal form, the base stands the same, but we find the fifth root of the number.
• The power being negative initially is just a signal to convert into the reciprocal to make computation easier.
• After computing the positive power and simplifying through the reciprocal, as shown, ending with the final simplification to value \(-2\).

Thus, negative exponents just mean "take the reciprocal and then apply." With these steps clearly laid out, resolving expressions with negative exponents becomes less daunting and more of a structured process.