Answer:
In our ordinary division, we are clear about the division of a number by the same number. The quotient is always 1 in such case.
For example:-
Divide 4 by 4.
Here,
[tex] \frac{4}{4} = 1[/tex]
Now let solve the above examples by using the laws of indices.
[tex] \frac{4}{4} = {4}^{1 - 1} = {4}^{0} = 1[/tex]
Similarly,
[tex] \frac{ {x}^{m} }{ {x}^{m} } = {x}^{m - m} = {x}^{0} = 1[/tex]
Thus,any base with zero index is equal to 1.
We need to tell the index of x for which x will be 1 , First of all , always remember that index of a number means power of the number or to which power it's raised to, like in 2³ , 2 is raised to 3 so 3 is the index. Also we knows an identity i.e
Let's put , m = n
[tex]{:\implies \quad \sf \dfrac{a^m}{a^m}=a^{m-m}}[/tex]
[tex]{:\implies \quad \boxed{\bf{a^{0}=1}}}[/tex]
Also , you should note that for a = 0 , the given expression becomes 0⁰ ,which is not defined , so the above expression is true only for [tex]{\bf{a\in (-\infty,0)\cup (0,\infty)}}[/tex]
Now , as we can replace variables , so , if we replace a by x , we get x⁰ = 1 .
Hence , the required index is 0
