Which of the following is equivalent to $36^{-\frac{1}{2}}$ ?
The objective is to evaluate the expression \(36^{-\frac{1}{2}}\) and select the correct option from the given choices: \(-18\), \(-6\), \(\frac{1}{18}\), and \(\frac{1}{6}\). The expression involves a base of \(36\) and a rational exponent of \(-\frac{1}{2}\). To solve this, we will apply standard exponent rules.
The rule for negative exponents states that for any non-zero real number \(a\) and any exponent \(n\), \(a^{-n} = \frac{1}{a^n}\). Applying this to our expression:
\[36^{-\frac{1}{2}} = \frac{1}{36^{\frac{1}{2}}}.\]
The rule for fractional exponents states that for any non-negative real number \(a\), \(a^{\frac{1}{n}} = \sqrt[n]{a}\). For the denominator \(36^{\frac{1}{2}}\), we set \(n=2\), which gives the square root of 36:
\[36^{\frac{1}{2}} = \sqrt{36}.\]
Since \(6 \times 6 = 36\), the square root of 36 is 6. Therefore, \(36^{\frac{1}{2}} = 6\).
Now that we have evaluated the denominator, we can substitute it back into our fraction from step 2:
\[36^{-\frac{1}{2}} = \frac{1}{36^{\frac{1}{2}}} = \frac{1}{6}.\]
Comparing this result to the provided options, we see that it matches the last option. Thus, \(\frac{1}{6}\) is the correct choice.
Understanding negative and fractional exponents is crucial in various fields, such as finance and physics. For example, in compound interest formulas, fractional exponents help calculate interest over periods shorter than a year. Similarly, in physics, the inverse-square law, which describes how forces like gravity or light intensity diminish with distance, often uses negative exponents (e.g., \(d^{-2}\)). Mastering these operations allows for more precise modeling of real-world phenomena.