Question

How do you convert $0.000098$ into scientific notation?

Answer 1

Hint: The scientific notation is a conventional way of representing a decimal number in the form of $a\times {{10}^{b}}$, where $a$ is a decimal number greater than or equal to one and less than ten, while $b$ is an integer which is an exponent over ten. Therefore for converting the given decimal number $0.000098$ into the scientific notation, we have to multiply it with ten repeatedly until it becomes equal to $9.8$. The number of times this multiplication is carried out will be equal to the negative of the exponent over ten in its scientific notation.

Complete step by step solution:
Let us write the given number in the above question as
$\Rightarrow n=0.000098$
We know that the scientific notation form of a number is expressed as $a\times {{10}^{b}}$, where $a$ is a decimal number greater than or equal to one and less than ten, and $b$ is an integer. This means that we have to multiply the given number repeatedly by ten until it becomes equal to $9.8$. Therefore, we multiply the both sides of the above equation by ${{10}^{5}}$ to get
$\begin{align}
  & \Rightarrow {{10}^{5}}n=0.000098\times {{10}^{5}} \\
 & \Rightarrow {{10}^{5}}n=9.8 \\
\end{align}$
Finally, dividing both the sides by \[{{10}^{5}}\], we get
\[\begin{align}
  & \Rightarrow \dfrac{{{10}^{5}}n}{{{10}^{5}}}=\dfrac{9.8}{{{10}^{5}}} \\
 & \Rightarrow n=9.8\times {{10}^{-5}} \\
\end{align}\]
Hence, the given decimal number is converted into the scientific notation and is written as \[9.8\times {{10}^{-5}}\].

Note: We can note that the absolute value of the power over ten in the scientific notation \[9.8\times {{10}^{-5}}\], which is equal to five, is also equal to the number of zeroes in the given decimal number $0.000098$. So we can make this as a checkpoint as well as a trick that in the scientific notation for a decimal number less than one, the exponent over ten will be equal to the negative of the number of zeroes encountered before the first non-zero digit in the given decimal number.