Question

Of the numbers $0.16,\sqrt {0.16} ,{\left( {0.16} \right)^2},0.1\overline 6 $ the least number is:
A. ${\left( {0.16} \right)^2}$
B. $\sqrt {0.16} $
C. $0.16$
D. $0.1\overline 6 $

Answer 1

Hint: It is a simple question where we need to find the smallest number. It can be done by calculating the individual values which are easier to find and then we can see which is the least one.
We must know what $0.1\overline 6 $ means because if we don’t know its meaning then we cannot compare all the values which are given. So here the bar sign on $6$ means that this digit keeps on repeating which means $0.1\overline 6 = 0.16666666......$
When we have the term $\dfrac{1}{3} = 0.33333333......$then we see that the term never finishes and here $3$ keeps on repeating so we can write this as $0.\overline 3 $

Complete step-by-step answer:
Here we are given the four numbers which are $0.16,\sqrt {0.16} ,{\left( {0.16} \right)^2},0.1\overline 6 $ and we need to find the least number out of these four numbers. So we can compare all four by making then convert into a simpler form which is in decimal digits.
So first we must know what $0.1\overline 6 $ means because if we don’t know its meaning then we cannot compare all the values which are given. So here the bar sign on $6$ means that this digit keeps on repeating which means $0.1\overline 6 = 0.16666666......$
So we get one value which is
$0.1\overline 6 = 0.16666666......$$ - - - - - (1)$
We have one number which is simply $0.16$$ - - - - - (2)$
Now we also have ${\left( {0.16} \right)^2}$
We can write ${\left( {0.16} \right)^2}$ as $0.16 \times 0.16$
We also know $0.16 = \dfrac{{16}}{{100}}$
So we get ${\left( {0.16} \right)^2}$$ = \dfrac{{16}}{{100}} \times \dfrac{{16}}{{100}} = \dfrac{{256}}{{10000}} = 0.0256$$ - - - - - - (3)$
Now we have $\sqrt {0.16} = \sqrt {\dfrac{{16}}{{100}}} = \dfrac{{\sqrt {16} }}{{\sqrt {100} }} = \dfrac{4}{{10}} = 0.4$$ - - - - - - (4)$
Now we have four values which are $0.16666.....,0.16,0.0256,0.4$
Out of these we know that $0.0256$ is the least one. So we can say that ${\left( {0.16} \right)^2}$ is the least out of the four numbers which are given.
Hence option A is correct.

Note: This question can also be done if we have the knowledge of the square roots and squares of the numbers which lies between $0{\text{ and 1}}$ in which we must know that:
(1) Square root of any number between $0{\text{ and 1}}$ is always more than that number.
(2) Square of the number between $0{\text{ and 1}}$ is always less than the number.
For example: ${\left( {0.16} \right)^2}$$ = \dfrac{{16}}{{100}} \times \dfrac{{16}}{{100}} = \dfrac{{256}}{{10000}} = 0.0256$
Here $0.0256 < 0.16$
And also we calculated $\sqrt {0.16} = \sqrt {\dfrac{{16}}{{100}}} = \dfrac{{\sqrt {16} }}{{\sqrt {100} }} = \dfrac{4}{{10}} = 0.4$
Here $0.4 > 0.16$
So by doing this we will get that out of square root of$0.16$, square of$0.16$and $0.16$ square of the number is least. So in this way also we can calculate our least number easily.