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If $2^4 - 2^3 = 2^3$ and $2^5 - 2^4 = 2^4$, then is below a rule of subtracting exponents with similar base and exponents which are just $1$ away from each other?

$$A^e - A^{e-1} = A^{e-1}$$

I will also like to get an visual intuition of why this works. Thanks.

Théophile
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LoveWithMaths
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    It works in the first case because $2^4=2^{3+1}=2.2^3$. In general, it is not true that $y^x=2y^{x-1}$ –  Sep 21 '18 at 03:39
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    No, what works in general is $,A^e - A^{e-1}=A \cdot A^{e-1} - A^{e-1}=(A-1)A^{e-1},$ instead. – dxiv Sep 21 '18 at 03:43
  • It works in the first case because 2^4=2^(3+1)=2*2^3. Why it wont work for other case ? – LoveWithMaths Sep 21 '18 at 16:00

1 Answers1

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For $A \ne 0$ we have:

$A^e - A^{e-1} = A^{e-1} \iff A^e=2A^{e-1} \iff A^e=2 A^e A^{-1} \iff 1=2 A^{-1} \iff A=2$.

Fred
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