Question

How do you write the answer in scientific notation given $\left( 6.1\times {{10}^{5}} \right)\left( 2.2\times {{10}^{6}} \right)$?

Answer 1

Hint: The scientific notation of a number is expressed in the form of $a\times {{10}^{m}}$, where a is a number greater than or equal to one, and less than ten, and m is an integer. The number given in the above question, which is $\left( 6.1\times {{10}^{5}} \right)\left( 2.2\times {{10}^{6}} \right)$, is a multiplication of two number written in their respective scientific notations. For writing it in the scientific notation, we need to multiply the powers of ten together, and the numbers with each other. Then using the properties of exponents, we can simplify the number obtained. And finally on arranging the number such that the number multiplied with the power of ten is between one and ten, we will get the scientific notation of the given number.

Complete step by step answer:
The number given to us in the above question is equal to $\left( 6.1\times {{10}^{5}} \right)\left( 2.2\times {{10}^{6}} \right)$. Let this number be equal to n so that we can write the equation
$\Rightarrow n=\left( 6.1\times {{10}^{5}} \right)\left( 2.2\times {{10}^{6}} \right)$
Now, using the properties of the multiplication, we can write the above expression as
$\Rightarrow n=6.1\times 2.2\times {{10}^{5}}\times {{10}^{6}}$
On multiplying $6.1$ by $2.2$, we get \[13.42\]. Therefore, we can put \[6.1\times 2.2=13.42\] in the above equation to get
$\Rightarrow n=13.42\times {{10}^{5}}\times {{10}^{6}}$
Now, using the exponent property of the multiplication of the same bases, we have ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$. Therefore, we can put ${{10}^{5}}\times {{10}^{6}}={{10}^{5+6}}$ in the above equation to get
$\begin{align}
  & \Rightarrow n=13.42\times {{10}^{5+6}} \\
 & \Rightarrow n=13.42\times {{10}^{11}} \\
\end{align}$
We know that the scientific notation of a number is expressed in the form of $a\times {{10}^{m}}$, where $a$ is a number greater than or equal to one, and less than ten. But in the above number, we have $13.42$ which is greater than ten. Therefore, we divide and multiply the above number to get
$\begin{align}
  & \Rightarrow n=13.42\times {{10}^{11}}\times \dfrac{10}{10} \\
 & \Rightarrow n=\dfrac{13.42}{10}\times {{10}^{11}}\times 10 \\
 & \Rightarrow n=1.342\times {{10}^{12}} \\
\end{align}$

Hence, we have written the number given in the above question in the scientific notation as $1.342\times {{10}^{12}}$.

Note: Do not forget to multiply and divide ten by the number $13.42\times {{10}^{11}}$. The number multiplied with ten must be between one and ten but the number $13.42$ is greater than ten. So it is necessary to reduce it to the number less than ten.