Zero power Itself

Algebra Level 2

We all know the exponential identity,

$a^{-n} = \frac{1}{a^{n}} \\ a^{0} = 1$

So, what is the value of $0^0$ using any of above identities?

Undefined -1 0 1

1 solution

Viki Zeta
Jun 27, 2016

On finding how they got $a^0 = 1$

$a^0 = a^{n-n} = a^{n} \times a^{-n} = \frac{a^n}{a^n}$

So, now on setting any value for $n > 0$ , you get

$0^0 = \frac{0^n}{0^n} ; n > 0 \\ => 0^0 = \frac{0}{0} \\ \text{We all know that '0' divided by '0' is undetermined or undefined, so }0^0 \text{ is undefined too.}$

Wrong. $\dfrac{0}{0}$ is not undefined, it is indeterminate. These two terms have completely different meanings

Hung Woei Neoh - 4 years, 11 months ago

I'll give a simple answer for which $\frac{0}{0}$ is undefined as-well-as indeterminate.

$\frac{0}{0} = 0 \times \frac{1}{0} = 0 * undefined = undefined, 0$

Viki Zeta - 4 years, 11 months ago

0÷0 is undefined. Get a dictionary.

Viki Zeta - 4 years, 11 months ago

I did my research before I reported this question

Indeterminate: An expression which has a value, but whose value cannot be determined exactly. For example, we know that

$a \times 0 = 0$

$a$ can take any value, therefore $\dfrac{0}{0}$ can actually take any value, and therefore is indeterminate

Undefined: An expression which cannot be defined or has no meaning at all. For example, consider

$a \times 0 = b$

We know $b$ cannot be non-zero. Therefore, $\dfrac{b}{0}$ , where $b\neq 0$ is undefined

Hung Woei Neoh - 4 years, 11 months ago

Let me come to your point of view,

$0^0 = 0^{-0} = \frac{1}{0^{0}}$ , according to your statement, $0^0$ is indeterminate, let it be, but $\frac{1}{0}$ is not defined, so the answer is undefined.

Viki Zeta - 4 years, 11 months ago

@Viki Zeta – Here, you are assuming that $\dfrac{1}{0^0} = \dfrac{1}{0}$ , which means you are saying that $0^0 = 0$

Hung Woei Neoh - 4 years, 11 months ago

@Hung Woei Neoh – No I mean any value maybe for $0^0$ , Even 0, but we know $\frac{1}{0}$ is undefined, so the answer is undefined.

Viki Zeta - 4 years, 11 months ago

@Viki Zeta – So what happens if $0^0$ is not zero? Your logic is flawed

Therefore, as long as we cannot determine the exact value , we say that it is indeterminate

Hung Woei Neoh - 4 years, 11 months ago

Zero power Itself

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