Question

question
which expression is equivalent to \\(\frac{3^{-4}}{3}\\)?

answer
\\(3^5\\) \\(\frac{1}{3^5}\\) \\(3^4\\) \\(3^3\\)

Explanation:

Step1: Recall exponent rule for division

When dividing exponents with the same base, we use the rule \( \frac{a^m}{a^n} = a^{m - n} \). Here, the base \( a = 3 \), \( m=-4 \), and \( n = 1 \) (since \( 3 = 3^1 \)).
So, \( \frac{3^{-4}}{3^1}=3^{-4 - 1} \)

Step2: Simplify the exponent

Calculate \( -4-1=-5 \), so we get \( 3^{-5} \).

Step3: Recall negative exponent rule

The rule \( a^{-n}=\frac{1}{a^n} \) applies here. So, \( 3^{-5}=\frac{1}{3^5} \).

Answer:

\( \frac{1}{3^5} \) (the option corresponding to \( \frac{1}{3^5} \) in the choices)