Question

How do you simplify and write $\dfrac{1.0\times {{10}^{-15}}}{4.2\times {{10}^{-7}}}$ in scientific notation?

Answer 1

Hint: In this question, we have to convert a fractional decimal notation into scientific notation. As we know, a decimal notation is a number that has a decimal point, which means the decimal can be removed by placing the numbers after the decimal in the denominator. Also, a scientific notation is either a very large number or a small number; it is represented when a number lies between 1 to 10 is multiplied by a power, which implies $a\times {{10}^{b}}$ where a is the number between 1 and 10 and b is either negative for small numbers. So, in this problem, we first split the multiplication into two fractional numbers. Then, we will solve them separately to get a new form of scientific notation, which is our required solution to the problem.

Complete step by step solution:
According to the question, we have to find the scientific notation from a fractional decimal notation.
The decimal notation given to us is $\dfrac{1.0\times {{10}^{-15}}}{4.2\times {{10}^{-7}}}$ -------- (1)
So, first we will split the equation (1) with respect to multiplication, to get two new fractional decimal number, we get
$\dfrac{1.0}{4.2}\times \dfrac{{{10}^{-15}}}{{{10}^{-7}}}$
Now, we will count the number of zeros after the decimal point of decimal number 1.0 and 4.2, we have
\[\begin{align}
  & \text{1}\text{.0 and 4}.2 \\
 & \text{ }\uparrow \text{ }\uparrow \\
\end{align}\]
So, we get 1 number after a decimal point and before 1 and 4. Therefore, we will divide 10 on both the numerator and the denominator in the first fraction number, we get
$\dfrac{\dfrac{10}{10}}{\dfrac{42}{10}}\times \dfrac{{{10}^{-15}}}{{{10}^{-7}}}$
On further simplification, we get
$\dfrac{5}{21}\times \dfrac{{{10}^{-15}}}{{{10}^{-7}}}$
Now, on dividing 5 by 21, we get the quotient equal to 0.238, therefore we get
$0.238\times \dfrac{{{10}^{-15}}}{{{10}^{-7}}}$
Now, we will apply the exponent rule $\dfrac{{{a}^{5}}}{{{a}^{6}}}={{a}^{5-6}}$ in the above expression, we get
$0.238\times {{10}^{-15-(-7)}}$
On opening the brackets of the above expression, we get
$0.238\times {{10}^{-15+7}}$
So, on further simplification, we get
$0.238\times {{10}^{-8}}$
Now, we will shift 1 decimal to the right, thus one 10 will be divided to the above expression, we get
$\dfrac{2.38}{10}\times {{10}^{-8}}$
Now, we will again apply the exponent rule $\dfrac{{{a}^{5}}}{{{a}^{6}}}={{a}^{5-6}}$ in the above expression, we get
$2.38\times {{10}^{-8-(1)}}$
Therefore, we get
$2.38\times {{10}^{-9}}$

Therefore, for the fractional decimal notation $\dfrac{1.0\times {{10}^{-15}}}{4.2\times {{10}^{-7}}}$, its scientific notation is equal to $2.38\times {{10}^{-9}}$

Note: While solving this problem, keep in mind the difference between scientific notation and decimal notation. Also, a scientific notation is only represented for decimal numbers. Do all the steps carefully to avoid confusion and error.