Zero Power

Zero Power

Any base that has a power of zero has a value of one. it does not matter whether the base is a number or a pronumeral. If the power of zero, its value is one. We can show this by looking at the following example, which can be simplified using two different methods.

$\textbf{Method 1}$
$$ \large \begin{align} \displaystyle
5^4 \div 5^4 &= \dfrac{5 \times 5 \times 5 \times 5}{5 \times 5 \times 5 \times 5} \\
&= \dfrac{625}{625} \\
&= 1
\end{align} $$

$\textbf{Method 2}$
$$ \large \begin{align} \displaystyle
5^4 \div 5^4 &= \dfrac{5^4}{5^4} \\
&= 5^{4-4} \\
&= 5^0
\end{align} $$
Since the two results should be the same, $5^0$ must equal $1$.
Any base that has an index (power) of zero is equal to $1$.
$$ \large a^0 = 1$$

Example 1

Find the value of $12^0$.

$12^0 = 1$

Example 2

Find the value of $(15a^4b^5c^6)^0$.

$(15a^4b^5c^6)^0 = 0$

Example 3

Find the value of $5^0 + 5$.

\( \begin{align} \displaystyle
5^0 + 5 &= 1 + 5 \\
&= 6
\end{align} \)

Example 4

Simplify $4a^2b^0$.

\( \begin{align} \displaystyle
4a^2b^0 &= 4a^2 \times 1 \\
&= 4a^2
\end{align} \)

Example 5

Simplify $\dfrac{9x^4 \times 4x^7}{2x^6 \times 3x^5}$.

\( \begin{align} \displaystyle
\dfrac{9x^4 \times 4x^7}{2x^6 \times 3x^5} &= \dfrac{9 \times 4}{2 \times 3} \times \dfrac{x^4 \times x^7}{x^6 \times x^5} \\
&= 6 \times x^{4+7-6 -5} \\
&= 6 \times x^0 \\
&= 6 \times 1 \\
&= 6
\end{align} \)

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