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Question Number 1748 by Rasheed Ahmad last updated on 14/Sep/15
Why   a^0 =1 and  a^(−3) =(1/a^3 )
$${Why}\:\:\:{a}^{\mathrm{0}} =\mathrm{1}\:{and}\:\:{a}^{−\mathrm{3}} =\frac{\mathrm{1}}{{a}^{\mathrm{3}} } \\ $$
Answered by 123456 last updated on 14/Sep/15
(a^x /a^y )=a^(x−y) ,a≠0  so  x=y  a^0 =a^(x−x) =(a^x /a^x )=1  a^(−x) =a^(0−x) =(a^0 /a^x )=(1/a^x )
$$\frac{{a}^{{x}} }{{a}^{{y}} }={a}^{{x}−{y}} ,{a}\neq\mathrm{0} \\ $$$$\mathrm{so} \\ $$$${x}={y} \\ $$$${a}^{\mathrm{0}} ={a}^{{x}−{x}} =\frac{{a}^{{x}} }{{a}^{{x}} }=\mathrm{1} \\ $$$${a}^{−{x}} ={a}^{\mathrm{0}−{x}} =\frac{{a}^{\mathrm{0}} }{{a}^{{x}} }=\frac{\mathrm{1}}{{a}^{{x}} } \\ $$

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