# How do you simplify 5^0?

Feb 14, 2017

#### Explanation:

While it is easy to understand ${5}^{2}$ and ${5}^{5}$, which are $5 \times 5$ and $5 \times 5 \times 5 \times 5 \times 5$

it is not easy to understand ${5}^{0}$.

So let us try it different way.

What is ${5}^{5} \div {5}^{2}$?

To answer this let us use division as fraction ${5}^{5} / {5}^{2}$ and this is

$\frac{5 \times 5 \times 5 \times 5 \times 5}{5 \times 5}$

= $\frac{5 \times 5 \times 5 \times \cancel{5} \times \cancel{5}}{\cancel{5} \times \cancel{5}}$

= ${5}^{3}$ and we can express this as 5^((5-2) i.e.

5^5-:5^2=5^((5-2)

In fact similarly we can generalize it to say

a^m-:a^n=a^m/a^n=a^((m-n), which is an identity

What is ${a}^{0}$ then. Well we can use above identity and can interpret it as

a^0=a^((m-m), but RHS is ${a}^{m} \div {a}^{m}$ or ${a}^{m} / {a}^{m}$ i.e. $1$

Hence for any $a$, we have ${a}^{0} = 1$ and so too for $a = 5$

and ${5}^{0} = 1$