Question

How do you write $\dfrac{1.68\times {{10}^{-1}}}{3\times {{10}^{4}}}$ in standard form?

Answer 1

Hint: We first solve the equation $\dfrac{1.68\times {{10}^{-1}}}{3\times {{10}^{4}}}$. Then we explain the process of expressing indices in their actual form of decimal. We start with the decimal number $0.56$ and move it leftwards to multiply the new number 10 with ${{10}^{-5}}$. The power of the 10 needs to become 0 to transform into decimal form.

Complete step-by-step solution:
The purpose of scientific notation is for scientists to write very large, or very small, numbers with ease.
We first try to simplify the given fraction $\dfrac{1.68\times {{10}^{-1}}}{3\times {{10}^{4}}}$. We get
$\dfrac{1.68\times {{10}^{-1}}}{3\times {{10}^{4}}}=\dfrac{1.68}{3}\times \dfrac{{{10}^{-1}}}{{{10}^{4}}}=0.56\times {{10}^{-5}}$.
We need to convert $0.56\times {{10}^{-5}}$ into standard form.
For the given number we move the decimal to the left side one position and bring one zero if there is no such number to cross each time. The decimal keeps going towards the left till the power of the 10 becomes 0 from ${{10}^{-5}}$. The more we move to the left, the more we multiply with 10.
We explain the first two steps. The decimal starts from its actual position in $0.56$.
Now it crosses the 0 in $0.56$ which means we have to multiply 10.
So, $0.56$ becomes $0.056$ and ${{10}^{-5}}$ becomes ${{10}^{-5}}\times 10={{10}^{-4}}$
Now in the second step the point crosses one more time bringing one extra zero. So, $0.056$ becomes $0.0056$ and ${{10}^{-4}}$ becomes ${{10}^{-4}}\times 10={{10}^{-3}}$.
The movement of the decimal point happens 5 times which means $0.56$ becomes $0.0000056$ and ${{10}^{-5}}$ becomes ${{10}^{-5}}\times {{10}^{5}}=1$.
Therefore, the standard form of $\dfrac{1.68\times {{10}^{-1}}}{3\times {{10}^{4}}}$ is $0.0000056$.

Note: We also can add the zeros after digit 6 after decimal. The use of zeroes is unnecessary. But in cases where we have digits other than 0 after decimal, we can’t ignore those digits. The final form will be $\dfrac{1.68\times {{10}^{-1}}}{3\times {{10}^{4}}}=0.56\times {{10}^{-5}}=0.0000056$.