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 $\dfrac{9x^4 \times 4x^7}{2x^6 \times 3x^5}$.
\( \begin{align} \displaystyle
\dfrac{9x^4 \times 4x^7}{2x^6 \times 3x^5} &= \dfrac{9 \times 4}{2 \times 3} \times \dfrac{x^4 \times x^7}{x^6 \times x^5} \\
&= 6 \times x^{4+7-6 -5} \\
&= 6 \times x^0 \\
&= 6 \times 1 \\
&= 6
\end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
