Question

What is the set of numbers to which $ \sqrt {10.24} $ belong?

Answer 1

Hint: We have to find the set of numbers to which the above given square root belongs, we will first solve the number inside the square root to see if it’s a surd or a perfect square rational number with a numerator and a denominator . If it can be expressed in the form of a numerator or denominator outside the square root sign we can say that the given number is a rational number , on the other hand if we cannot get the number to come out of the square root sign we will say that the given number is an irrational number and is a surd.

Complete step by step solution:
We will solve the number inside the square root sign . The number has a decimal so first we will convert the decimal back into fraction and then solve it from there.
 $\Rightarrow \sqrt {10.24} $ can be written as
 $ \Rightarrow \sqrt {\dfrac{{1024}}{{100}}} $
The numerator will be written as,
 $\Rightarrow \sqrt {\dfrac{{{2^{10}}}}{{{{10}^2}}}} $
The number on getting out of the square root sign will have all its power divided into half so the power $ 10 $ will become $ 5 $ and the power $ 2 $ will become $ 1 $
So we get
 $\Rightarrow \dfrac{{{2^5}}}{{10}} $
 $\Rightarrow \dfrac{{32}}{{10}} $
since the number is written in the form of a fraction the given number can be said to belong to rational numbers.
So, the correct answer is “ $ \dfrac{{32}}{{10}} $ ”.

Note: A number is said to be a rational number if the given number can be written in the form of $ \dfrac{P}{Q} $ where both $ P{\text{ and }}Q $ are numbers belonging to the set of real numbers.