Question

which expression is equivalent to ((4^{4} \times 4^{-1})^{-4})?
answer
(\bigcirc \frac{1}{4^{8}}) (\bigcirc \frac{1}{4}) (\bigcirc \frac{1}{4^{20}}) (\bigcirc \frac{1}{4^{12}})

Explanation:

Step1: Use the exponent rule \(a^m \times a^n = a^{m + n}\) inside the parentheses.

For \(4^4 \times 4^{-1}\), we have \(m = 4\) and \(n=- 1\), so \(4^4\times4^{-1}=4^{4+( - 1)}=4^{3}\)

Step2: Use the exponent rule \((a^m)^n=a^{m\times n}\) on \((4^{3})^{-4}\)

Here \(m = 3\) and \(n=-4\), so \((4^{3})^{-4}=4^{3\times(-4)}=4^{-12}\)

Step3: Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \(4^{-12}=\frac{1}{4^{12}}\)

Answer:

\(\frac{1}{4^{12}}\) (corresponding to the option \(\frac{1}{4^{12}}\))