0 and 1 in exponents

Algebra Level 3

How many statement(s) below is/are correct?

Statement 1 : 1 x = 1 1^x = 1 for all real values of x x .
Statement 2 : x 1 = 1 x^1 = 1 for all real values of x x .
Statement 3 : 0 x = 0 0^x = 0 for all real values of x x .
Statement 4 : x 0 = 1 x^0 = 1 for all real values of x x .

0 1 2 3 4

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1 solution

Hung Woei Neoh
May 5, 2016

Relevant wiki: Rules of Exponents - Algebraic

Only the first statement is correct. It is true that 1 x = 1 1^x = 1 for all real values of x x .

Here are what the other statements should have been, and the reasons why they are not true:

2) x 1 = 1 x^1 = 1 only holds true when x = 1 x=1 . Example: If x = 2 , x 1 = 2 1 = 2 1 x=2,\;x^1 = 2^1 = 2 \neq 1 . The true statement should be

x 1 = x x^1 = x for all real values of x x .

3) 0 x = 0 0^x = 0 does not hold true for any non-positive value of x x . Example: x = 1 , 0 x = 0 1 = 1 0 = Undefined 0 x=-1,\;0^x = 0^{-1} = \dfrac{1}{0} = \text{Undefined} \neq 0 . The true statement should be

0 x = 0 0^x = 0 for all positive real values of x x , that is, x > 0 x>0 .

4) x 0 = 1 x^0 = 1 holds true for all real values of x x except x = 0 x=0 . The value of 0 0 0^0 is indeterminate. The true statement should be

x 0 = 1 x^0 = 1 for all real values of x , x 0 x, x \neq 0 .

Here's an interesting article on 0 0 0^0

Thanks for sharing the problem along with an awesome solution. You've explained all the four statements very clearly. Keep it up.

Special thanks for adding the links into the solution. :-)

@Hung Woei Neoh

Sandeep Bhardwaj - 5 years, 1 month ago

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The original solution was not like this. As far as I know, at least 2 mods or admins edited my solution, and I further added a bit more after that

Hung Woei Neoh - 5 years, 1 month ago

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We were featuring your problem, and decided to add more explanation / clarity to your statements. Thanks for adding in more details :)

Calvin Lin Staff - 5 years, 1 month ago

Yeah, I guess that's cool. However, you can learn from the edits made by mods/admins to improve further. Finally the solution looks amusing to me. ;)

Sandeep Bhardwaj - 5 years, 1 month ago

Haha, did thw naughty staff play a role in this XD

Ashish Menon - 5 years, 1 month ago

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@Ashish Menon The original solution only contains the corrected statements. A few edits were made by mods and admins, and I found that the explanation as to why the statements were wrong were insufficient. So I edited them XD

Hung Woei Neoh - 5 years, 1 month ago

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@Hung Woei Neoh That is true but did you understand who I was referring to as naughty staff :P

Ashish Menon - 5 years, 1 month ago

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@Ashish Menon Nope. I'm still new here

Hung Woei Neoh - 5 years, 1 month ago

Uno elevado a infinito es indeterminado.

One raised to infinity is undetermined.

1⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹......................infinite..............................⁹⁹⁹⁹⁹ is undetermined.

ma pm - 5 years, 1 month ago

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If I remember correctly, all real values of x x meant:

< x < -\infty < x < \infty or ( , ) (-\infty,\infty)

It does not include infinity

Hung Woei Neoh - 5 years, 1 month ago

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Can you add "real values" into the problem? You can edit your problem through the menu.

Calvin Lin Staff - 5 years, 1 month ago

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@Calvin Lin Ah, right, my mistake. I've edited the problem and solution accordingly. Thanks for notifying.

Hung Woei Neoh - 5 years, 1 month ago

I think this question is good :)

Ashish Menon - 5 years, 1 month ago

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Thank you!

Hung Woei Neoh - 5 years, 1 month ago

Edit your second statement please , It is x 1 = 1 And not x 1 = x \text {It is }x^1=1 \text{ And not } x^1=x

Sabhrant Sachan - 5 years, 1 month ago

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Hmm? I asked how many statements were correct, right?

Hung Woei Neoh - 5 years, 1 month ago

If you were talking about my solution, then you should know that I'm giving out the correction to the false statements.

x 1 = x x^1 = x for all values of x x . For example, 2 1 = 2 2^1=2 or 3 1 = 3 3^1=3

Hung Woei Neoh - 5 years, 1 month ago

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I've added the clarifying statement of "Here are what the corrected statements should say:" to your solution.

Calvin Lin Staff - 5 years, 1 month ago

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@Calvin Lin Thank you!

Hung Woei Neoh - 5 years, 1 month ago

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