Question

What should be the index of x so that the value of x so that it's value will be equal to 1?

Answer 1

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,

$\frac{4}{4} = 1$

Now let solve the above examples by using the laws of indices.

$\frac{4}{4} = {4}^{1 - 1} = {4}^{0} = 1$

Similarly,

$\frac{ {x}^{m} }{ {x}^{m} } = {x}^{m - m} = {x}^{0} = 1$

Thus,any base with zero index is equal to 1.

Answer 2

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

• ${\boxed{\bf{\dfrac{a^m}{a^n}=a^{m-n}}}}$

Let's put , m = n

${:\implies \quad \sf \dfrac{a^m}{a^m}=a^{m-m}}$

${:\implies \quad \boxed{\bf{a^{0}=1}}}$

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 ${\bf{a\in (-\infty,0)\cup (0,\infty)}}$

Now , as we can replace variables , so , if we replace a by x , we get x⁰ = 1 .

Hence , the required index is 0