Question

Explain or demonstrate why the product rule of exponents, bxby = bx+y is true forpositive integer values of X and y

Answer 1

b raised to the power x can be written as,

$b^x=(b\times b\times b\times b\ldots\ldots.xtimes)\ldots\ldots(1)$

b raised to the power y can be written as,

$b^y=(b\times b\times b\times b\ldots\ldots.ytimes)\ldots\ldots\ldots(2)$

Multiplying both the equation (1) and (2),

$\begin{gathered} b^xb^y=(b\times b\times b\ldots\ldots.xtimes)(b\times b\times b\ldots\ldots..ytimes) \\ =(b\times b\times b\ldots\ldots..(x+y)times) \\ =b^{(x+y)} \end{gathered}$

Thus, when the two exponents with the same base is multiplied then their powers are added.