Question

Solve: \[0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3}\]
A) \[0.1\]
B) \[0.111\]
C) \[0.1211\]
D) \[0.2341\]
E) \[0.3\]

Answer 1

Hint:
Here, we will first convert these decimal numbers into fractions. Then, we will take the LCM of these fractions and simplify it. Finally, converting the fractional answer into a decimal number will help us to find the required answer.

Formula Used:
\[{\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}\]

Complete step by step solution:
Now, in order to find the answer to the sum of these three decimals raised to some powers respectively, we will first convert the given decimals into fractions.
As we can see, all three decimal numbers are having the decimal point after one digit from the unit’s place respectively.
Since, the decimal point is after 1 digit, hence, after converting these decimal numbers to fractions, we will have only 1 zero in the respective denominators.
Thus, \[0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3}\] can be written as:
\[0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = \dfrac{1}{{10}} + {\left( {\dfrac{1}{{10}}} \right)^2} + {\left( {\dfrac{1}{{10}}} \right)^3}\]
Now, using the formula, \[{\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}\], we get
We can further write our answer as:
\[ \Rightarrow 0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = \dfrac{1}{{10}} + \dfrac{1}{{{{10}^2}}} + \dfrac{1}{{{{10}^3}}}\]
Now, we know that while doing the square of the number 10, we have 2 zeros in total and after doing the cube, we have 3 zeros and so on.
Apply in the exponent on the terms, we get
\[ \Rightarrow 0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = \dfrac{1}{{10}} + \dfrac{1}{{100}} + \dfrac{1}{{1000}}\]
Now, taking the LCM, we get
\[ \Rightarrow 0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = \dfrac{{100 + 10 + 1}}{{1000}} = \dfrac{{111}}{{1000}}\]
Since the denominator is having 3 zeros; hence, to convert this fraction as a decimal, we will put a zeros after three digits from the unit’s place.
 \[ \Rightarrow 0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = 0.111\]

Therefore, option B is the correct answer.

Note:
An alternate way of solving this question is to directly solve using the decimals.
Since, we are required to find the value of: \[0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3}\]
If the decimal number is having a power 2, then, we transfer the decimal point towards the left by one more digit. Similarly, if the power is 3, then, we transfer the decimal point towards 2 more digits to the left.
Hence, we get,
\[0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = 0.1 + 0.01 + 0.001\]
Now, adding all these numbers by keeping the decimal sign at the same place, we get,
\[ \Rightarrow 0.1 + {\left( {0.1} \right)^2} + {\left( {0.1} \right)^3} = 0.111\]
Therefore, option B is the correct answer.