What is the value of $\left(-\frac{1}{2}\right)^{-4} ?$
The objective is to compute the value of the expression \(\left(-\frac{1}{2}\right)^{-4}\). The expression involves a base of \(-\frac{1}{2}\) and an exponent of \(-4\). To solve this, we will use fundamental laws of exponents: the negative exponent rule, which states \(x^{-n} = \frac{1}{x^n}\) for \(x eq 0\), and the property regarding the parity of exponents with negative bases, which states \((-x)^n = x^n\) when \(n\) is an even integer.
To simplify the expression \(\left(-\frac{1}{2}\right)^{-4}\), we first apply the negative exponent rule \(x^{-n} = \frac{1}{x^n}\):
\[\left(-\frac{1}{2}\right)^{-4} = \frac{1}{\left(-\frac{1}{2}\right)^4}.\]
Now, we focus on evaluating the denominator \(\left(-\frac{1}{2}\right)^4\). Since the exponent \(4\) is an even integer, any negative number raised to the power of \(4\) becomes positive. Thus, we have:
\[\left(-\frac{1}{2}\right)^4 = \left(\frac{1}{2}\right)^4.\]
Next, we distribute the exponent to both the numerator and the denominator:
\[\left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1}{16}.\]
Now that we have determined the value of the denominator, we can complete the calculation by taking the reciprocal:
\[\frac{1}{\left(-\frac{1}{2}\right)^4} = \frac{1}{\frac{1}{16}} = 16.\]
Therefore, the value of the expression \(\left(-\frac{1}{2}\right)^{-4}\) is \(16\), which corresponds to option a.
In fields like electronics or computer science, scientific notation and exponents are frequently used to express very small or very large values, such as capacitor tolerances or memory capacities. For example, if a signal is scaled by a factor of \(\left(0.5\right)^{-4}\), it is actually being amplified by a factor of \(16\), as \(0.5 = \frac{1}{2}\) and \((\frac{1}{2})^{-4} = 2^4 = 16\). Mastering these exponent rules is essential for scaling data and understanding signals in digital systems.