Question

The value of \[\dfrac{1}{\sqrt[5]{e}}\], up to four decimal places is \['p'\]. Find the approximate value of \[10p\].

Answer 1

Hint: We solve this problem first by taking the value of \['e'\] as 2.1783. Then we find the value of \[\dfrac{1}{\sqrt[5]{e}}\] by substituting the value of \['e'\] up to four decimal places. Then we use the simple rules of rounding off numbers to get the value of \[10p\]. The rules of rounding off are
(i) If the successor decimal place digit is less than or equal to ‘5’ then the number is unchanged.
(ii) If the successor decimal place digit is greater than ‘5’ then the number is increased by ‘1’

Complete step by step answer:
We are given that the value of \[\dfrac{1}{\sqrt[5]{e}}\], up to four decimal places is \['p'\].
So, we can take
\[\Rightarrow p=\dfrac{1}{\sqrt[5]{e}}\]
We know that the value of \['e'\] up to four decimal places is 2.7183
By substituting the value of \['e'\] in above equation we get
\[\Rightarrow p=\dfrac{1}{\sqrt[5]{2.7183}}\]
Now by taking the denominator to numerator we get’
\[\begin{align}
  & \Rightarrow p={{\left( 2.1753 \right)}^{-\dfrac{1}{5}}} \\
 & \Rightarrow p={{\left( 2.7183 \right)}^{-0.2}} \\
 & \Rightarrow p=0.8187 \\
\end{align}\]
Here, we can see the value of \['p'\] up to four decimal places.
Now, let us use the rounding off numbers rules to find the required result.
We know that the rules of rounding off numbers are
(i) If the successor decimal place digit is less than or equal to ‘5’ then the number is unchanged.
(ii) If the successor decimal place digit is greater than ‘5’ then the number is increased by ‘1’
Now, let us round off the number \['p'\] to three decimal places.
Here, we can see that the successor decimal digit of third decimal place of \['p'\] is ‘7’ which is greater than ‘5’ so we need to add one to the number.
By using the above result we get
\[\Rightarrow p=0.819\]
Now, let us round off to two decimal places.
Here, we can see that the successor decimal digit of second decimal place of \['p'\] is ‘9’ which is greater than ‘5’ so we need to add one to the number.
By using the above result we get
\[\Rightarrow p=0.82\]
Now, let us round off to one decimal place.
Here, we can see that the successor decimal digit of the second decimal place of \['p'\] is ‘2’ which is less than ‘5’ so we need to leave the number as it is.
By using the above result we get
\[\Rightarrow p=0.8\]
Now, by multiplying the above equation with 12 on both sides we get
\[\Rightarrow 10p=8\]

Therefore, the value of \[10p\] is 8.

Note: Students will make mistakes in the rounding off rules only. At the rounding off of decimal places to two decimal places while increasing the digit by ‘1’ we get\[p=0.82\]. But students may take as
\[p=0.81\], because the number successor to ‘1’ is ‘9’. This gives the wrong answer. So rounding off the decimal digits is an important task in this question.