Question

State true or false.
Charge of an electron \[0.000,000,000,000,000,000,16coulomb \]is equal to \[1.6 \times {10^{ - 19}}coulomb\]

Answer 1

Hint:In this question we convert the decimal number into fraction form and then collecting the powers of ten we write it in the desired form and check if it is equal to the value given in the question.
* A number in decimal form can be converted into fraction by writing the number in numerator without the decimal and counting the digits after the decimal which helps us to write decimal which is one along with that many zeroes.
Example \[x.yz = \dfrac{{xyz}}{{100}}\] here two digits after the decimal so we write hundred in the denominator.

Complete step-by-step answer:
We are given charge of an electron as \[0.000,000,000,000,000,000,16coulomb\]
Looking at the value given above we can tell there are total \[20\] digits after the decimal.
Writing the number in its fraction form, the numerator has the number without the decimal and the denominator has one along with twenty zeroes i.e.
\[0.000,000,000,000,000,000,16coulomb = \dfrac{{0000,000,000,000,000,000,16}}{{100000000000000000000}}coulomb\]
Since we know \[100000000000000000000 = \underbrace {10 \times 10 \times 10.... \times 10}_{20times}\]
And we know that \[\underbrace {a \times a..... \times a}_n = {a^n}\]
Here \[a = 10,n = 20\] so we can write
\[100000000000000000000 = {10^{20}}\]
Also any number of zeroes before any digit don’t count like in the numerator so we can remove them and we can write the term in the numerator as
\[0000,000,000,000,000,000,16 = 16\]
Substituting the values of numerator and denominator back in the fraction
\[0.000,000,000,000,000,000,16coulomb = \dfrac{{16}}{{{{10}^{20}}}}coulomb\]
Now we can write \[{10^{20}} = 10 \times {10^{19}}\] { when the base is same powers are added}
\[0.000,000,000,000,000,000,16coulomb = \dfrac{{16}}{{10 \times {{10}^{19}}}}coulomb\]
Now when we divide the value \[16\] by \[10\] we get \[1.6\] or we just place the decimal after one digit moving from the right side to left side.
Therefore, \[0.000,000,000,000,000,000,16coulomb = \dfrac{{1.6}}{{{{10}^{19}}}}coulomb\]
Now we can multiply both numerator and denominator by \[{10^{ - 19}}\]
\[0.000,000,000,000,000,000,16coulomb = \dfrac{{1.6}}{{{{10}^{19}}}} \times \dfrac{{{{10}^{ - 19}}}}{{{{10}^{ - 19}}}}coulomb\]
Since base is same powers can be added in the denominator
\[
  0.000,000,000,000,000,000,16coulomb = \dfrac{{1.6 \times {{10}^{ - 19}}}}{{{{10}^{19 - 19}}}}coulomb \\
  0.000,000,000,000,000,000,16coulomb = \dfrac{{1.6 \times {{10}^{ - 19}}}}{{{{10}^0}}}coulomb \\
 \]
Since we know any number to power zero becomes one. Therefore, \[{10^0} = 1\]
\[0.000,000,000,000,000,000,16coulomb = \dfrac{{1.6 \times {{10}^{ - 19}}}}{1}coulomb\]
So, \[0.000,000,000,000,000,000,16coulomb\] can be written as \[1.6 \times {10^{ - 19}}coulomb\]
Thus, the statement is TRUE.

Note:Students are likely to make calculation mistakes in counting the number of zeroes to be put in the denominator and they can very easily get manipulated with so many zeroes before the number in the numerator.