Question

Which of the following numbers 0.1, 0.11, ${\left( {0.11} \right)^2}$ and $\sqrt {0.0001} $ is the greatest?

Answer 1

Hint: Here, we are given 4 numbers and we need to find out the greatest number amongst them. For comparing these numbers, we first need to convert all the four given numbers into decimal form first. After conversion, we can easily compare them.

Complete step by step solution:
In this question, we are given four numbers and we need to find out which of these 4 numbers is the greatest.
The given numbers are: 0.1, 0.11, ${\left( {0.11} \right)^2}$ and $\sqrt {0.0001} $
Now, to compare these numbers, we first need to convert all four numbers into decimal form first.
Let us evaluate the value of all these 4 numbers one by one separately.
First let us take $\sqrt {0.0001} $
Now, to find the value of $\sqrt {0.0001} $, let us use the log method. Let $\sqrt {0.0001} $ be equal to x. Therefore,
\[
   \Rightarrow x = \sqrt {0.0001} \\
   \Rightarrow x = {\left( {0.0001} \right)^{\dfrac{1}{2}}} \\
   \Rightarrow \log x = \log {\left( {0.0001} \right)^{\dfrac{1}{2}}} \\
   \Rightarrow \log x = \dfrac{1}{2}\log \left( {0.0001} \right) \\
   \Rightarrow \log x = \dfrac{1}{2}\left( { - 4} \right) \\
   \Rightarrow \log x = - 2 \\
   \Rightarrow x = anit\log \left( { - 2} \right) \\
   \Rightarrow x = 0.01 \\
 \]
Hence, $\sqrt {0.0001} = 0.01$
Now, let us take ${\left( {0.11} \right)^2}$.
Now, to find the value of ${\left( {0.11} \right)^2}$, simply multiply 0.11 with 0.11. Therefore, we get
$ \Rightarrow {\left( {0.11} \right)^2} = 0.11 \times 0.11 = 0.0121$
Next number is 0.11.
Now, we can write 0.11 as 0.110. Therefore,
$ \Rightarrow 0.11 = 0.110$
And the last number is 0.1
Now, we can also write 0.1 as 0.100. Therefore,
$ \Rightarrow 0.1 = 0.100$
Therefore, now we have our numbers as 0.100, 0.110, 0.012 and 0.010.
Observing all these 4 numbers, we can arrange them in descending order.
$0.110 > 0.100 > 0.012 > 0.010$.
Therefore, out of 0.1, 0.11, ${\left( {0.11} \right)^2}$ and $\sqrt {0.0001} $ 0.11 is the greatest number.

Note:
Note that here for finding out the greatest number, converting all the numbers into decimal form first is very important. Also, you can add as many zeros as you want at the end or at the start of any decimal number, but not in middle of the decimal number.