Question

Product of $ 0.01\times 0.001\times 0.0001=\_\_\_\_\_\_. $

Answer 1

Hint: Multiply the numbers without any leading 0s and put as many decimal points in the final product as the sum of the number of decimal points of the factors (the numbers which were multiplied).
The number of decimal points in $ 0.01 $ , $ 0.001 $ and $ 0.0001 $ are 2, 3 and 4 respectively.
$ 0.\underbrace{001}_{\begin{smallmatrix} \text{ 3} \\ \text{decimal} \\ \text{ points} \end{smallmatrix}} $
The numbers $ 0.01 $, $ 0.001 $ and $ 0.0001 $ without any decimal points are $ 1 $ , $ 1 $ and $ 1 $ respectively.

Complete step-by-step answer:
If we consider the numbers without the decimal point and leading 0s, we get $ 1 $ , $ 1 $ and $ 1 $ respectively for the numbers $ 0.01 $, $ 0.001 $ and $ 0.0001 $ .
So, we consider the product $ 1\times 1\times 1 $ which is equal to 1.
Now, for the position of the decimal point, consider the number of decimal points in $ 0.01 $, $ 0.001 $ and $ 0.0001 $ which are 2, 3 and 4 respectively.

Therefore, we need $ 2+3+4=9 $ decimal places in the final answer, including the product 1. This can be written as $ 0.\underbrace{000000001}_{\text{9 decimal points}} $ , which is the required answer.

Note: We can also represent the numbers in the scientific form and then multiply using rules of exponents.
 $ 1\times {{10}^{-2}}\times 1\times {{10}^{-3}}\times 1\times {{10}^{-4}}=1\times {{10}^{-9}} $
The decimal point shifts to the right when multiplying by 10, and shifts to the left when divided by 10.
If we get trailing 0s in the product, they MUST BE considered when placing the decimal point.
e.g. $ 0.25\times 0.04=0.\underbrace{0100}_{\begin{smallmatrix} \text{4 places} \\ \text{including} \\ \text{100} \end{smallmatrix}} $