\(\bullet\) What does \(x^0\) signify ?

\(\bullet\) What is the actual meaning of \(0!\) ?

\(\bullet\) What does \(x^0\) signify ?

\(\bullet\) What is the actual meaning of \(0!\) ?

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TopNewest\(0!\) means the number of permutations of 0 distinct objects, which is nothing!

The way of permuting 0 objects is not doing anything, which is 1 way!

Therefore, \(0! = 1\) – Samuraiwarm Tsunayoshi · 2 years, 3 months ago

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Well , \(x^{0}\) means

xis raised to the power of0.Now moving on to your second question ,

\[ n! = n\times (n-1)! \\ (n-1)! = \dfrac{n!}{n}\]

Input

n=1to get0!=1.But what you have asked is the significance of

0!right ?Actually it has no significance of its own , we have just alloted it an arbitrary value of one , so that we may use it for our benefits . One of it's applications is in the Choose function .

\[ \binom{12}{0} = \dfrac{12!}{(12)!(0)!}\]

This way , one can even incorporate

0!into calculations ignoring the fact that \(\dfrac{1}{0}\) is undefined .If you want , you can read this , though this might be slightly off topic .

@Sandeep Bhardwaj A bit of expert's advice may be needed here since I'm not satisfied with my reply . Pls help him out sir .

Thanks :) – Azhaghu Roopesh M · 2 years, 3 months ago

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I hope, it will give you what you're looking for. If you've any doubts, you can ask me here.

Thanks! – Sandeep Bhardwaj · 2 years, 3 months ago

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– Abhijeet Verma · 2 years, 3 months ago

Thanks you sir.Log in to reply

– Azhaghu Roopesh M · 2 years, 3 months ago

As usual ,your interpretations are the best :DLog in to reply

– Sandeep Bhardwaj · 2 years, 3 months ago

Thank you very much. \(\ddot \smile\).Log in to reply

– Abhijeet Verma · 2 years, 3 months ago

Thanks, but what is the actual meaning of "something is raised to the power of 0" =1 ? (I don't want the proof )Log in to reply

Provided \(x \neq 0\)

Is this satisfactory explanation? – Krishna Sharma · 2 years, 3 months ago

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xxx. x^2=xx. x^0=? – Abhijeet Verma · 2 years, 3 months agoLog in to reply

A simple analogy would be subtraction. At first, we are taught the rules of subtraction like this. \( 7 - 3 \) is like having \(7\) apples and taking \(3\) away. But \( 3-7 \) isn't defined according to that explanation.Or maybe \( (2+ 3i) - (4 + 7i) \). How would you explain \( (2+ 3i) - (4 + 7i) \). By extending the rules of subtraction beyond natural numbers, we lose the ability to

explainthe equations but at the same time gain the ability to use it in more scenarios.Another example might be multiplication. \( 2 \times 3 \) is defined as repeated addition,i.e, \( 2 + 2 + 2 \). What about \( \pi \times e \)? You can't explain it like that. Same case as above. We extend the scope of multiplication beyond natural numbers but lose the ability to "explain" it. – Siddhartha Srivastava · 2 years, 3 months ago

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– Abhijeet Verma · 2 years, 3 months ago

OK, Thank you sir.Log in to reply

– Azhaghu Roopesh M · 2 years, 3 months ago

As usual your explanations are too good :)Log in to reply

– Krishna Sharma · 2 years, 3 months ago

You can write \(x^0 = \dfrac{x}{x}\) or any power of x i.e \(\dfrac{x^n}{x^n}\) \(n \in \mathbb R\)Log in to reply

– Azhaghu Roopesh M · 2 years, 3 months ago

Actually I might just not be able to convincingly explain it out to you , so can you wait till Sandeep sir replies ?Log in to reply

– Abhijeet Verma · 2 years, 3 months ago

surely ,thanks.Log in to reply

– Azhaghu Roopesh M · 2 years, 3 months ago

No , not at all ! Why thank me ? I didn't help you out at all .Log in to reply

– Abhijeet Verma · 2 years, 3 months ago

Thanks for finding time to add your comment. I think you will be cracking JEE this year , right?Log in to reply

– Azhaghu Roopesh M · 2 years, 3 months ago

Well , I'll try to crack it . It's not that easy , you know .Log in to reply

– Abhijeet Verma · 2 years, 3 months ago

Best of luck for your examLog in to reply

– Azhaghu Roopesh M · 2 years, 3 months ago

Thanks :DLog in to reply