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}) (\frac{2^5}{2^9})

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 \(\frac{4^{11}}{4^{13}} = 4^{11-13}=4^{-2}=\frac{1}{4^2}=\frac{1}{16}\).
• For \(\frac{1}{16}\): It is already in simplified form, and we know \(4^{-2}=\frac{1}{16}\), so this is equivalent to \(4^{-2}\).
• For \(\frac{4}{4^2}\): Simplify \(\frac{4}{4^2}=\frac{4^1}{4^2}=4^{1 - 2}=4^{-1}=\frac{1}{4}\), which is not equal to \(4^{-2}\) or \(\frac{1}{16}\).
• For \(4^{-2}\): By definition, \(4^{-2}=\frac{1}{4^2}=\frac{1}{16}\).
• For \(\frac{2^5}{2^9}\): Use the rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{2^5}{2^9}=2^{5 - 9}=2^{-4}=\frac{1}{2^4}=\frac{1}{16}\) (since \(2^4 = 16\)).

Step2: Identify the odd one out.

We see that \(\frac{4}{4^2}\) simplifies to \(\frac{1}{4}\), while the others simplify to \(\frac{1}{16}\) (or are equal to \(4^{-2}\) which is \(\frac{1}{16}\)).

Answer:

\(\boldsymbol{\frac{4}{4^2}}\) (because \(\frac{4^{11}}{4^{13}} = 4^{-2}=\frac{1}{16}\), \(\frac{1}{16}\) is equal to \(4^{-2}\), \(\frac{2^5}{2^9}=2^{-4}=\frac{1}{16}\), but \(\frac{4}{4^2}=4^{-1}=\frac{1}{4}\) which is different from \(\frac{1}{16}\))