Question

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

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^{n}} \) for any non - zero 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

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

Step4: Substitute back into the expression

Since \( 36^{\frac{1}{2}} = 6 \), then \( \frac{1}{36^{\frac{1}{2}}}=\frac{1}{6} \).

Answer:

\(\frac{1}{6}\) (the first option in the given choices)