An identity that expresses a rule for multiplying the quantities in exponential notation is called the product rule of exponents.
The numbers are multiplied directly but it is not possible to multiply the quantities in exponential notation. The multiplication of exponential form quantities requires special multiplication law to find their product. In mathematics, two types of exponential form quantities involve in multiplication. Hence, the following two properties are the formulas with proofs for multiplying the exponential quantities.
The product of exponents with same base is equal to the sum of their powers with same base.
$b^{\displaystyle m} \times b^{\displaystyle n} \,=\, b^{\displaystyle \,m+n}$
The product of same exponents with different bases is equal to the power of a product of their bases.
$b^{\displaystyle m} \times c^{\displaystyle m} \,=\, (b \times c)^{\displaystyle m}$
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