Product Law of Exponents with same Exponent

Formula

$\large b^{\displaystyle m} \times c^{\displaystyle m} = (b \times c)^{\displaystyle m}$

Proof

$b$ and $c$ are two literals. Multiply the literal $b$ by same literal number $m$ times to represent the product them and it is $b^{\displaystyle m}$. Similarly, multiply the literal $c$ by the same literal $m$ times to represent their product and it is $c^{\displaystyle m}$.

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

$c^{\displaystyle m} = \underbrace{c \times c \times c \ldots c}_{\displaystyle m \, factors}$

Multiply both exponential terms to obtain the product of them.

$b^{\displaystyle m} \times c^{\displaystyle m}$ $\,=\,$ $\underbrace{b \times b \times b \ldots b}_{\displaystyle m \, factors}$ $\times$ $\underbrace{c \times c \times c \ldots c}_{\displaystyle m \, factors}$

Multiply each literal $b$ by every literal number $c$ to write product of them.

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

In this case, the total number of multiplying factors of $b$ is $m$ and also the total number of multiplicative factors of $c$ is also $m$. Hence, the total number of multiplying factors of product of $b$ and $c$ is also $m$. It can be written in exponential notation as follows.

$b^{\displaystyle m} \times c^{\displaystyle m} = (b \times c)^{\displaystyle m}$

The mathematical proof expresses that the product of two exponential terms having same exponent is equal to the product of bases of the terms, raised to the same exponent. It is called the product law of exponents or indices.

The product formula of exponents is not limited to two terms and it can be applied to several exponential terms. Hence, the product property of the indices can be written in general form.

$b^{\displaystyle m} \times c^{\displaystyle m} \times d^{\displaystyle m} \ldots$ $\,=\,$ $(b \times c \times d \ldots)^{\displaystyle m}$

Verification

$2^4$ and $3^4$ are two exponential terms and they both have same exponent $4$.

$2^4 = 2 \times 2 \times 2 \times 2 = 16$

$3^4 = 3 \times 3 \times 3 \times 3 = 81$

Multiply both of them.

$2^4 \times 3^4 = 16 \times 81 = 1296$

Now, multiply both bases and find the value of the product of them raised to the power of $4$.

${(2 \times 3)}^4 = 6^4$ $=$ $6 \times 6 \times 6 \times 6 = 1296$

Now, check both values to understand this property.

$\therefore \,\,\,\,\,\,$ $2^4 \times 3^4 \,=\, {(2 \times 3)}^4 \,=\, 1296$

The product rule of exponents is successfully verified mathematically in numerical system. Hence, the product law of indices is also called as the product identity of the exponential terms.

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