$b^{\displaystyle m} \times b^{\displaystyle n} \,=\, b^{\displaystyle \,m+n}$

The product of multiplication of exponents with the same base is equal to the sum of their powers with same base, is called the product rule of exponents with same base.

In mathematics, two or more exponents with the same base are involved in multiplication but it is not possible to multiply them directly same as the numbers. So, a special product law is required for multiplying the powers with the same base.

There is a property for multiplying the indices with same base and it reveals that the product of multiplication of two or more exponents with the same base can be obtained by adding the exponents with the same base.

$b^{\displaystyle m} \times b^{\displaystyle n} \times b^{\displaystyle o} \ldots$ $\,=\,$ $b^{\displaystyle m+n+o \cdots}$

The product law of powers with same base is used in two different cases.

- The product of exponents with same base is simplified as the sum of the exponents with the same base.
- The addition of exponents with a base is expanded as the product of the exponents with the same base.

Learn how to prove the product rule of indices with same base in mathematics.

The product of two numbers $16$ and $64$ is $1024$.

$16 \times 64 \,\,=\,\, 1024$

Express, $16$, $64$ and $1024$ in exponential notation on the basis of number $4$.

$(1)\,\,\,\,\,\,$ $16 = 4 \times 4 = 4^2$

$(2)\,\,\,\,\,\,$ $64 = 4 \times 4 \times 4 = 4^3$

$(3)\,\,\,\,\,\,$ $1024 = 4 \times 4 \times 4 \times 4 \times 4 = 4^5$

Now, write the mathematical relationship between $16$, $64$ and $1024$ in the form of exponents with same base.

$16 \times 64 \,\,=\,\, 1024$

$\implies 4^2 \times 4^3 \,\,=\,\, 4^5$

Observe the exponents of the three exponential terms, it clears that the product of exponents with the same base can be obtained by adding the exponents with the same base.

$\therefore \,\,\,\,\,\, 4^2 \times 4^3 \,\,=\,\, 4^{2\,+\,3}$

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