Power Rule of Zero Indices

Formula

$b^{\displaystyle 0} = 1$

zero exponent power rule

The value of an exponential term having zero as its power is one. Exponential terms are appeared with zero exponents in mathematics but the value of them is same as its coefficient. Therefore, the value of any quantity consists of zero as its power is one.

The property is related to the power of the exponential terms. So, the power rule of zero exponent is called the power law of exponential term having zero as its exponent.

Proof

$b^{\displaystyle 0}$ is an exponential term. The meaning of term $b^{\displaystyle 0}$ is one time of $b^{\displaystyle 0}$.

$b^{\displaystyle 0} = 1 \times b^{\displaystyle 0}$

The term $b^{\displaystyle 0}$ has zero as its power. The real meaning of the exponent zero is zero times the literal quantity is multiplied by itself. So, no need to write the literal quantity $b$ at least once in product form.

Therefore, the value of the exponential term $b^{\displaystyle 0}$ is equal to its coefficient.

$b^{\displaystyle 0} = 1$

Example

$8^{\displaystyle 0}$ is an exponential term, having $8$ as base and zero as exponent.

$\implies 8^{\displaystyle 0} = 1 \times 8^{\displaystyle 0} $

$\implies 8^{\displaystyle 0} = 1$

Not only the value of $8^{\displaystyle 0}$, the value of any number which contains zero as its exponent is always one.

This power rule of zero exponent is basically called as an identity but it is in algebraic form. For this reason, it is known as an algebraic identity. However, this algebraic identity is in exponential form. Therefore, it is also called as an exponential identity.

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