Question

How to solve this?
$\dfrac{\dfrac{6.02\times 10^{20}}{6.02\times 10^{23}}}{\dfrac{100}{1000}}$

Answer 1

Hint: We are given a fraction of fractions. For this, we will first solve the fractions separately in the numerator and the denominator and then we will calculate the final result. To solve this, we should have the knowledge of how to solve powers in fractions and some rules using which this will be solved.

Complete step-by-step solution:
We should know that for any integers $a$ and $b$ and for any number $x$, the following holds true:
$\dfrac{x^a}{x^b}=x^{a-b}$
We will use this formula here:
We have in the numerator:
$\dfrac{6.02\times {{10}^{20}}}{6.02\times {{10}^{23}}}$
The term of 6.02 will be cancelled out from both numerator and denominator. Now, we only have the powers of 10 which will be solved using the formula given above. So we have:
$\dfrac{10^{20}}{10^{23}}=10^{20-23}=10^{-3}$
So the fraction in the numerator becomes:
$\Rightarrow \dfrac{6.02\times {{10}^{20}}}{6.02\times {{10}^{23}}}={{10}^{-3}}_{^{^{{}}}}$
Now, we solve the denominator:
$\Rightarrow \dfrac{100}{1000}=\dfrac{1}{10}=10^{-1}$
Hence, the fraction in the denominator becomes:
$\Rightarrow \dfrac{100}{1000}=10^{-1}$
Hence, the whole fraction becomes:
$\Rightarrow \dfrac{\dfrac{6.02\times {{10}^{20}}}{6.02\times {{10}^{23}}}}{\dfrac{100}{1000}}=\dfrac{{{10}^{-3}}}{{{10}^{-1}}}$
Again we apply the same formula of powers and we get:
$\Rightarrow \dfrac{\dfrac{6.02\times {{10}^{20}}}{6.02\times {{10}^{23}}}}{\dfrac{100}{1000}}={{10}^{-2}}$
Hence the answer has been found out.

Note: If your calculations are fast and accurate you can easily do the division without solving the numerator and denominator separately. You can directly cut off powers from numerator and denominator and then take the denominator of the denominator and put it in the denominator which is how division works in fraction. To simplify it further you can write $100=10^2$ and $1000=10^3$ and make it easier.