Question

select all the expressions that are equivalent to \\(\left(4^{-1}\
ight)^{-8}\\).
\\(\frac{1}{4^{-9}}\\) \\(4^{8}\\)
\\(\frac{1}{4^{-8}}\\) \\(4^{-9}\\)

Explanation:

Step1: Simplify \((4^{-1})^{-8}\)

Using the exponent rule \((a^m)^n = a^{m\times n}\), we have \(4^{(-1)\times(-8)} = 4^{8}\).

Step2: Analyze \(\frac{1}{4^{-9}}\)

Using the rule \(a^{-n}=\frac{1}{a^n}\), \(\frac{1}{4^{-9}} = 4^{9}\), which is not equal to \(4^{8}\), so this is incorrect.

Step3: Analyze \(4^{8}\)

This is the result we got from Step1, so it is equivalent.

Step4: Analyze \(\frac{1}{4^{-8}}\)

Using \(a^{-n}=\frac{1}{a^n}\), \(\frac{1}{4^{-8}} = 4^{8}\), so this is equivalent.

Step5: Analyze \(4^{-9}\)

This is \(4^{-9}\), which is not equal to \(4^{8}\), so this is incorrect.

Answer:

\(4^{8}\), \(\frac{1}{4^{-8}}\)