Question

How do you write $3.5\times {{10}^{6}}$ in standard form?

Answer 1

Hint: To write the scientific form into standard form we will multiply the given number with ${{10}^{-1}}$ or divide the given number with $10$. Now we will shift the decimal one place to the right side of the digit. Now we will apply the exponential rule and simplify the value. Again, we will do the same process of multiplying ${{10}^{-1}}$ or dividing with $10$ until we will get the power of the $10$ as $0$. i.e., we will continue with the process until we will get no powers in the given number.

Complete step-by-step answer:
Given that $3.5\times {{10}^{6}}$.
Multiplying the given number with ${{10}^{-1}}$ and moving the decimal value to one-digit right side, then we will get
 $3.5\times {{10}^{6}}=35.0\times {{10}^{6}}\times {{10}^{-1}}$
We have the exponential rule ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, then
$\begin{align}
  & 3.5\times {{10}^{6}}=35.0\times {{10}^{6-1}} \\
 & \Rightarrow 3.5\times {{10}^{6}}=35.0\times {{10}^{5}} \\
\end{align}$
Again, multiplying the above number with ${{10}^{-1}}$ an moving the decimal value to one-digit right side, then we will get
$35.0\times {{10}^{5}}=350.0\times {{10}^{5}}\times {{10}^{-1}}$
We have the exponential rule ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, then
$\begin{align}
  & 35.0\times {{10}^{5}}=350.0\times {{10}^{5-1}} \\
 & \Rightarrow 35.0\times {{10}^{5}}=350.0\times {{10}^{4}} \\
\end{align}$
Again, multiplying the above number with ${{10}^{-1}}$ an moving the decimal value to one-digit right side, then we will get
$350.0\times {{10}^{4}}=3500.0\times {{10}^{5}}\times {{10}^{-1}}$
We have the exponential rule ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, then
$\begin{align}
  & 350.0\times {{10}^{4}}=3500.0\times {{10}^{4-1}} \\
 & \Rightarrow 350.0\times {{10}^{4}}=3500.0\times {{10}^{3}} \\
\end{align}$
Again, multiplying the above number with ${{10}^{-1}}$ an moving the decimal value to one-digit right side, then we will get
$3500.0\times {{10}^{3}}=35000.0\times {{10}^{3}}\times {{10}^{-1}}$
We have the exponential rule ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, then
$\begin{align}
  & 3500.0\times {{10}^{3}}=35000.0\times {{10}^{3-1}} \\
 & \Rightarrow 3500.0\times {{10}^{3}}=35000.0\times {{10}^{2}} \\
\end{align}$
Again, multiplying the above number with ${{10}^{-1}}$ an moving the decimal value to one-digit right side, then we will get
$35000.0\times {{10}^{2}}=350000.0\times {{10}^{2}}\times {{10}^{-1}}$
We have the exponential rule ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, then
$\begin{align}
  & 35000.0\times {{10}^{2}}=350000.0\times {{10}^{2-1}} \\
 & \Rightarrow 35000.0\times {{10}^{2}}=350000.0\times {{10}^{1}} \\
\end{align}$
Again, multiplying the above number with ${{10}^{-1}}$ an moving the decimal value to one-digit right side, then we will get
$350000.0\times {{10}^{1}}=3500000.0\times {{10}^{1}}\times {{10}^{-1}}$
We have the exponential rule ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, then
$\begin{align}
  & 350000.0\times {{10}^{1}}=3500000.0\times {{10}^{1-1}} \\
 & \Rightarrow 350000.0\times {{10}^{1}}=3500000.0\times {{10}^{0}} \\
\end{align}$
We know that the value of ${{a}^{0}}=1$, then
$3.5\times {{10}^{6}}=3500000$

Note: In the problem we have used the law which says that the zero placed after the decimal has no value. Based on this law we have used the zero after the decimal wherever it is useful. But we should consider that the zero which is placed before the decimal has a value, so don’t neglect the zero before the decimal.