Question

which expression is equivalent to ((3^{-4})^{6})? answer (\bigcirc) (3^{24}) (\bigcirc) (3^{10}) (\bigcirc) (3^{2}) (\bigcirc) (\frac{1}{3^{24}})

Explanation:

Step1: Recall the power of a power rule

The power of a power rule states that \((a^m)^n = a^{m\times n}\). Here, \(a = 3\), \(m=- 4\) and \(n = 6\).
So, for \((3^{-4})^{6}\), we multiply the exponents: \(-4\times6=-24\). Thus, \((3^{-4})^{6}=3^{-24}\).

Step2: Recall the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) (where \(a
eq0\) and \(n\) is an integer).
Applying this rule to \(3^{-24}\), we get \(3^{-24}=\frac{1}{3^{24}}\).

Answer:

\(\frac{1}{3^{24}}\) (the option \(\boldsymbol{\frac{1}{3^{24}}}\))