Question

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

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) for any non - zero real number \(a\) and positive integer \(n\). So, for \(36^{-\frac{1}{2}}\), we can rewrite it as \(\frac{1}{36^{\frac{1}{2}}}\).

Step2: Recall the definition of fractional exponents

The fractional exponent rule states that \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). When \(m = 1\), \(a^{\frac{1}{n}}=\sqrt[n]{a}\). So, \(36^{\frac{1}{2}}=\sqrt{36}\).

Step3: Calculate the square root of 36

We know that \(\sqrt{36} = 6\) because \(6\times6=36\).

Step4: Substitute the value back

Since \(36^{\frac{1}{2}} = 6\), then \(\frac{1}{36^{\frac{1}{2}}}=\frac{1}{6}\)? Wait, no, wait. Wait, \(36^{\frac{1}{2}}=\sqrt{36} = 6\), so \(\frac{1}{36^{\frac{1}{2}}}=\frac{1}{6}\)? Wait, no, wait, let's check again. Wait, \(36^{-\frac{1}{2}}=\frac{1}{36^{\frac{1}{2}}}=\frac{1}{\sqrt{36}}=\frac{1}{6}\)? Wait, but let's check the options. Wait, the options are \(- 18\), \(-6\), \(\frac{1}{18}\), \(\frac{1}{6}\). Wait, did I make a mistake? Wait, no, \(36^{\frac{1}{2}}=\sqrt{36} = 6\), so \(36^{-\frac{1}{2}}=\frac{1}{36^{\frac{1}{2}}}=\frac{1}{6}\).

Answer:

\(\frac{1}{6}\) (the option corresponding to \(\frac{1}{6}\))