# How do you simplify 1/2^0?

Mar 27, 2016

$\frac{1}{2} ^ 0 = \textcolor{g r e e n}{1}$

#### Explanation:

Any (non-zero) number to the power of zero equals $1$
So $\frac{1}{\textcolor{red}{{2}^{0}}} = \frac{1}{\textcolor{red}{1}} = 1$

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What is the logic behind "any number to the power of zero equals $1$"?

${a}^{n} = {a}^{n - 1} \times a$
$\textcolor{w h i t e}{\text{XXX}}$that is $a$ multiplied by itself $n$ times
$\textcolor{w h i t e}{\text{XXX}}$is the same as
$\textcolor{w h i t e}{\text{XXXXXX}} a$ multiplied together $\left(n - 1\right)$ times
$\textcolor{w h i t e}{\text{XXXXXX}}$and then multiplied by $a$ one more time.

This could be re-written as
${a}^{n - 1} = {a}^{n} \div a$ (provided $a \ne 0$)
So
$\textcolor{w h i t e}{\text{XXX}} {a}^{2} = {a}^{3} \div a$
$\textcolor{w h i t e}{\text{XXX}} {a}^{1} = {a}^{2} \div a$
$\textcolor{w h i t e}{\text{XXX}} {a}^{0} = {a}^{1} \div a$
but
$\textcolor{w h i t e}{\text{XXX}} {a}^{1} = a$
and therefore
$\textcolor{w h i t e}{\text{XXX}} {a}^{0} = a \div a = 1$ (again, provided $a \ne 0$)