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My son mistaken to answer his math question: $2^{-x} = 0.5^{x}$ He said this must be false, I asked ChatGpt but still not satisfied with the answer because I want to explain to my son at age 12.

The strange part is: when you look at $2^{-x}$ the result of Y must be negative while $0.5^x$ is positive.

J. W. Tanner
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Akam
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2 Answers2

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Hint:

$$a^{-b} = \dfrac {1}{a^b} = \bigg ( \dfrac {1}{a} \bigg )^b$$ See also Robert Israel's comment.

bjcolby15
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The main part of $2^{-n}$ is to understand $2^{-1}.$ $a^{-1}$ is generally the solution of the equation $a\cdot x=1,$ i.e. the inverse element to $a$ with respect to multiplication. $x=a^{-1}$ is how we write this element. Hence, $2^{-1}$ solves $2\cdot x =1$ which makes $2^{-1}=\dfrac{1}{2}.$ Finally, $$ 2^{-n}=\left(2^{-1}\right)^n=\left(\dfrac{1}{2}\right)^n=0.5^n=\dfrac{1}{2^n}>0 $$

Marius S.L.
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  • Thanks for nice explanation, can you provide example for n = -1,-2? – Akam Jan 23 '24 at 17:40
  • If $n<0$ then $-n>0$ and we get $2^{-n}=\underbrace{2\cdot\ldots\cdot 2}_{-n\text{ times}}.$ If you meant examples like $2^{-3}$ then we have $2^{-3}=\left(\dfrac{1}{2}\right)^3=\dfrac{1}{2}\cdot\dfrac{1}{2}\cdot\dfrac{1}{2}=\dfrac{1}{8}.$ – Marius S.L. Jan 23 '24 at 17:59