How do you multiply scientific notation $(0.33 \times 10^{-3}) \times (0.2 \times 10)$ ?

Hint:The given question is related to the concept of scientific notation. To solve this question, we will use the concept of scientific notation as well as the rules of the exponents. Here, in this question we will first rewrite the expression, then use the rule of exponent to multiply the $10s$ terms. At the end, we will write the expression obtained in the scientific notation.

Complete step by step answer:
Scientific notation is a means of expressing very large or small numbers by powers of ten so that the values are much more easily understood. These are used in expressing elemental or atomic quantities, cosmic or space quantities and technological metrics. Given is $(0.33 \times 10^{-3}) \times (0.2 \times 10)$. We start solving this question, by first rewriting the given expression as:
$\Rightarrow \left( {0.33 \times 0.2} \right) \times \left( {{{10}^{ - 3}} \times 10} \right) \\
\Rightarrow 0.066 \times \left( {{{10}^{ - 3}} \times {{10}^1}} \right) \\ $
Next, we will use the following rule of exponent to multiply the $10s$ terms.
$ \Rightarrow {x^a} \times {x^b} = {x^{a + b}}$
Using the above written rule of exponent, we get,
$\Rightarrow 0.066 \times \left( {{{10}^{ - 3}} \times {{10}^1}} \right) \\
\Rightarrow 0.066 \times \left( {{{10}^{ - 3 + 1}}} \right) \\
\Rightarrow 0.066 \times \left( {{{10}^{ - 2}}} \right) \\ $
Now, we will write this expression in scientific notation. For doing that we will move the decimal point two places to the left meaning, we need to subtract two from the $10s$ exponent.
$\Rightarrow 0.066 \times \left( {{{10}^{ - 2 - 2}}} \right) \\
\therefore 6.6 \times \left( {{{10}^{ - 4}}} \right) \\ $
Therefore, our required answer is $6.6 \times \left( {{{10}^{ - 4}}} \right)$.

Note:The given question could only be solved by those who know the concept of scientific notation and exponents very clearly. Here, in the given question we used a rule of exponent, students should remember all the basic rules of exponents. The common mistake which students tend to make is in calculation. Along with that, they also forget to convert the expression back into the scientific notation.