Answer: The required value of k is -1.
Step-by-step explanation: We are given to find the value of k in the product of the following powers :
[tex]10^{-3}.10.10^k=10^{-3}~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We will be using the following properties of exponents to solve the given equation.
[tex](i)~a^b.a^c=a^{b+c},\\\\(ii)~a^x=a^y~~~~\Rightarrow x=y.[/tex]
From equation (i) we have
[tex]10^{-3}.10.10^k=10^{-3}\\\\\Rightarrow 10^{-3+1+k}=10^{-3}\\\\\Rightarrow 10^{-2+k}=10^{-3}\\\\\Rightarrow -2+k=-3\\\\\Rightarrow k=-3+2\\\\\Rightarrow k=-1[/tex]
Therefore, the required value of k is -1.
