Question

1. which one does not belong? circle your answer and explain your reasoning. (\frac{4^{11}}{4^{13}}) (\frac{1}{16}) (\frac{4}{4^2}) (4^{-2})

Explanation:

Step1: Simplify each expression using exponent rules.

For \(\frac{4^{11}}{4^{13}}\), use the rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{4^{11}}{4^{13}} = 4^{11-13}=4^{-2}\).
For \(\frac{1}{16}\), note that \(16 = 4^2\), so \(\frac{1}{16}=\frac{1}{4^2}=4^{-2}\) (since \(a^{-n}=\frac{1}{a^n}\)).
For \(\frac{4}{4^2}\), use the rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{4}{4^2}=4^{1 - 2}=4^{-1}\).
For \(4^{-2}\), it is already in simplified form as \(4^{-2}\).

Step2: Compare the simplified forms.

We see that \(\frac{4^{11}}{4^{13}}\), \(\frac{1}{16}\), and \(4^{-2}\) all simplify to \(4^{-2}\), while \(\frac{4}{4^2}\) simplifies to \(4^{-1}\), which is different.

Answer:

The expression that does not belong is \(\boldsymbol{\frac{4}{4^2}}\) (or equivalently \(4^{-1}\)) because the other three expressions simplify to \(4^{-2}\) (or \(\frac{1}{16}\) or \(\frac{4^{11}}{4^{13}}\)) and this one simplifies to \(4^{-1}\).