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 :
\(10^{-3}.10.10^k=10^{-3}~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\)
We will be using the following properties of exponents to solve the given equation.
\((i)~a^b.a^c=a^{b+c},\\\\(ii)~a^x=a^y~~~~\Rightarrow x=y.\)
From equation (i) we have
\(10^{-3}.10.10^k=10^{-3}\\\\\Rightarrow 10^{-3+1+k}=10^{-3}\\\\\Rightarrow 10^{-2+k}=10^{-3}\\\\\Rightarrow -2+k=-3\\\\\Rightarrow k=-3+2\\\\\Rightarrow k=-1\)
Therefore, the required value of k is -1.
