Question

which of the following is equivalent to
$36^{-\frac{1}{2}}$?
options:
-18
-6
$\frac{1}{18}$
$\frac{1}{6}$

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) (where \(a
eq0\) and \(n\) is a positive integer). Also, for a number with a fractional exponent, \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). But here we have \(36^{-\frac{1}{2}}\), so first apply the negative exponent rule: \(36^{-\frac{1}{2}}=\frac{1}{36^{\frac{1}{2}}}\).

Step2: Evaluate \(36^{\frac{1}{2}}\)

The exponent \(\frac{1}{2}\) means taking the square root. So \(\sqrt{36} = 6\), since \(6\times6 = 36\).

Step3: Substitute back

Now substitute \(\sqrt{36}=6\) into \(\frac{1}{36^{\frac{1}{2}}}\), we get \(\frac{1}{6}\)? Wait, no, wait. Wait, \(36^{-\frac{1}{2}}=\frac{1}{36^{\frac{1}{2}}}=\frac{1}{\sqrt{36}}=\frac{1}{6}\)? Wait, but the options have \(\frac{1}{18}\), \(-6\), \(-18\), \(\frac{1}{6}\). Wait, maybe I misread the original expression. Wait, the original is \(36^{-\frac{1}{2}}\)? Wait, or is it \(-\frac{1}{36^{2}}\)? Wait, the image shows "36 -1/2" maybe? Wait, no, the user's image: "Which of the following is equivalent to \(36^{-\frac{1}{2}}\)?" Wait, the options are -18, -6, 1/18, 1/6. Wait, let's recalculate. \(36^{\frac{1}{2}}=\sqrt{36}=6\), so \(36^{-\frac{1}{2}}=\frac{1}{36^{\frac{1}{2}}}=\frac{1}{6}\). So the correct option is \(\frac{1}{6}\).

Answer:

\(\frac{1}{6}\) (assuming the option is labeled with \(\frac{1}{6}\), like the last option in the image)