Question

which expression is equivalent to \\(\frac{1}{6^3}\\)? (1 point)\
\\(\bigcirc\\ (6^{-1})^{-1}\\)\
\\(\bigcirc\\ 6^{-10} \times 6^8\\)\
\\(\bigcirc\\ 6^{-1} \times 6^{-2}\\)\
\\(\bigcirc\\ (6^{-4})^2\\)

Explanation:

Step1: Recall exponent rules

Recall that \(\frac{1}{a^n}=a^{-n}\), so \(\frac{1}{6^3}=6^{-3}\). Also, recall the power - of - a - power rule \((a^m)^n = a^{m\times n}\) and the product rule \(a^m\times a^n=a^{m + n}\).

Step2: Analyze Option 1: \((6^{-1})^{-1}\)

Using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), for \((6^{-1})^{-1}\), we have \(m=- 1\) and \(n = - 1\). So \((6^{-1})^{-1}=6^{(-1)\times(-1)}=6^{1}=6
eq6^{-3}\).

Step3: Analyze Option 2: \(6^{-10}\times6^{8}\)

Using the product rule \(a^m\times a^n=a^{m + n}\), here \(m=-10\) and \(n = 8\). So \(6^{-10}\times6^{8}=6^{-10 + 8}=6^{-2}
eq6^{-3}\).

Step4: Analyze Option 3: \(6^{-1}\times6^{-2}\)

Using the product rule \(a^m\times a^n=a^{m + n}\), here \(m=-1\) and \(n=-2\). So \(6^{-1}\times6^{-2}=6^{-1+( - 2)}=6^{-3}\), which is equal to \(\frac{1}{6^3}\).

Step5: Analyze Option 4: \((6^{-4})^{2}\)

Using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), here \(m = - 4\) and \(n=2\). So \((6^{-4})^{2}=6^{-4\times2}=6^{-8}
eq6^{-3}\).

Answer:

\(6^{-1}\times6^{-2}\) (the third option)