Power Rule of Exponent of an Exponential term

Formula

${(b^{\displaystyle m})}^{\displaystyle n} = b^{\displaystyle mn}$

power rule of exponent

The value of an exponential term which contains another power, is equal to the value of the exponential term whose exponent is the product of both powers of the term.

It is about the relation between the powers when a term contains more powers. Hence, the rule is called the power law of power of an exponential term. It is mainly used as a formula to obtain the quantity of an exponential term which consists of another power.

Proof

$b^{\displaystyle m}$ is an exponential term and formed by the product of $m$ times of literal quantity $b$.

$b^{\displaystyle m} = b \times b \times b \times \ldots \times b$

Assume, the exponential term $b^{\displaystyle m}$ is multiplied to itself $n$ times and the product of them is written as ${(b^{\displaystyle m})}^{\displaystyle n}$ in exponential notation.

${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{b^{\displaystyle m} \times b^{\displaystyle m} \times b^{\displaystyle m} \times \ldots \times b^{\displaystyle m}}_{\displaystyle n \, factors}$

Express each exponential term $b^{\displaystyle m}$ in its product form.

${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{\underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \ldots \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors}}_{\displaystyle n \, factors}$

Each $b^{\displaystyle m}$ term contains $m$ multiplying factors of $b$. So, $n$ terms contains $n \times m$ multiplying factors. Therefore, the number of multiplying factors is $mn$ in the product.

${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m \times n \, factors}$

$\implies {(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle mn \, factors}$

The literal quantity $b$ is multiplied by itself $mn$ times. So, it can be expressed in exponential form.

${(b^{\displaystyle m})}^{\displaystyle n} = b^{\displaystyle mn}$

Verification

${\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4}$ is an exponential term which contains another power. Use fundamental method to expand it for obtaining its value.

${\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4} = \Big(5^{\displaystyle 3}\Big) \times \Big(5^{\displaystyle 3}\Big) \times \Big(5^{\displaystyle 3}\Big) \times \Big(5^{\displaystyle 3}\Big) $

$\implies {\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4} = (5 \times 5 \times 5) \times (5 \times 5 \times 5) \times (5 \times 5 \times 5) \times (5 \times 5 \times 5)$

$\implies {\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4} = 125 \times 125 \times 125 \times 125$

$\implies {\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4} = 244140625$

The value of ${\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4}$ is $244140625$.

Now multiply both exponents and obtain the value.

$5^{\displaystyle (3 \times 4)} = 5^{\displaystyle 12}$

$\implies 5^{\displaystyle (3 \times 4)} = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5$

$\implies 5^{\displaystyle (3 \times 4)} = 244140625$

Therefore, ${\Big(5^{\displaystyle 3}\Big)}^{\displaystyle 4} = 5^{\displaystyle (3 \times 4)} = 244140625$

The power rule of power of an exponential term is verified in numerical system and it is true for all the values. Therefore, the power law is called as an identity but it is derived in algebraic form. Due to this reason, the identity can be called as an algebraic identity.

The algebraic identity is in terms of exponential terms. Hence, the power rule of exponent of an exponential term can be called as exponential identity in indices.

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