Question

which expressions are equivalent to \\(\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}\\)?
choose 2 answers:
a \\((5^{-2})^2\\)
b \\((5^{-4})^0\\)
c \\(\frac{5^1}{5^4}\\)
d \\(5^2 \cdot 5^{-6}\\)

Explanation:

Step1: Simplify the original expression

The original expression is \(\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}\). Using the rule of exponents \(a^m\cdot a^n=a^{m + n}\) and \(\frac{1}{a}=a^{-1}\), we can rewrite \(\frac{1}{5}\) as \(5^{-1}\). So the expression becomes \((5^{-1})^4 = 5^{-4}\) (using \((a^m)^n=a^{mn}\)).

Step2: Analyze Option A

For option A: \((5^{-2})^2\). Using the power - of - a - power rule \((a^m)^n=a^{mn}\), we have \((5^{-2})^2=5^{-2\times2}=5^{-4}\), which is equal to the simplified original expression.

Step3: Analyze Option B

For option B: \((5^{-4})^0\). Using the rule \(a^0 = 1\) (\(a
eq0\)), we get \((5^{-4})^0 = 1\), which is not equal to \(5^{-4}\).

Step4: Analyze Option C

For option C: \(\frac{5^{1}}{5^{4}}\). Using the quotient rule of exponents \(a^m\div a^n=a^{m - n}\), we have \(\frac{5^{1}}{5^{4}}=5^{1-4}=5^{-3}\), which is not equal to \(5^{-4}\). Wait, there is a mistake here. Wait, \(\frac{5^{1}}{5^{4}}=5^{1 - 4}=5^{-3}\)? No, wait, the original expression is \(5^{-4}\). Wait, maybe I made a mistake. Wait, \(\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}=\frac{1}{5^4}=5^{-4}\). And \(\frac{5^{1}}{5^{4}}=5^{1-4}=5^{-3}\)? No, that's wrong. Wait, \(\frac{5^{1}}{5^{4}}=\frac{5}{5\times5\times5\times5}=\frac{1}{5\times5\times5}=\frac{1}{5^3}=5^{-3}\). Wait, maybe I misread the option. Wait, the option C is \(\frac{5^{1}}{5^{4}}\)? Wait, no, maybe it's a typo? Wait, no, let's re - check. Wait, the original expression is \(5^{-4}\). Let's check option D.

Step5: Analyze Option D

For option D: \(5^{2}\cdot5^{-6}\). Using the product rule of exponents \(a^m\cdot a^n=a^{m + n}\), we have \(5^{2}\cdot5^{-6}=5^{2+( - 6)}=5^{-4}\), which is equal to the simplified original expression. And for option A, we had \((5^{-2})^2 = 5^{-4}\), option D: \(5^{2}\cdot5^{-6}=5^{-4}\), option B: \((5^{-4})^0 = 1\), option C: \(\frac{5^{1}}{5^{4}}=5^{-3}\). So the correct options are A and D.

Answer:

A. \((5^{-2})^{2}\)
D. \(5^{2}\cdot5^{-6}\)