Question

How do you find the square root of \[\dfrac{{26}}{{89}}\] ?

Answer 1

Hint: In the above question, we are given a fractional number \[\dfrac{{26}}{{89}}\] . We have to find the square root of the given rational number i.e. \[\sqrt {\dfrac{{26}}{{89}}} \] . Here, the numerator, \[26\] is a composite number as \[26\]can be written as \[26 = 2 \times 13\] but the denominator \[89\] is a prime number and hence can be written in multiplication of factors. Since both the numerator and the denominator do not have common factors and square factors, therefore the square root of the rational number \[\dfrac{{26}}{{89}}\] can not be written in simpler form and hence \[\sqrt {\dfrac{{26}}{{89}}} \] is an irrational number.

Complete step by step answer:
Given rational number is \[\dfrac{{26}}{{89}}\] .
We have to find \[\sqrt {\dfrac{{26}}{{89}}} \] .
Since the numerator and denominator here cannot be written in a simpler form, therefore \[\sqrt {\dfrac{{26}}{{89}}} \] is a irrational number and thus we can only write it into its rational approximations.
Now, the square root of \[\dfrac{{26}}{{89}}\] can also be written as the square root of \[26\]divided by the square root of \[89\] . You can divide the fraction inside the radical before or outside the radical after you calculate the square root. The result will be the same in both cases.
Let us first calculate the square root of the numerator and the square root of the denominator separately, and then we divide the two of them.
Now the square root of \[26\] can be calculated as,
\[ \Rightarrow \sqrt {26} = 5.099\]
Also, the square root of \[89\] can be calculated as,
\[ \Rightarrow \sqrt {89} = 9.434\]
Now \[\sqrt {\dfrac{{26}}{{89}}} \] can be written as
\[ \Rightarrow \sqrt {\dfrac{{26}}{{89}}} = \dfrac{{\sqrt {26} }}{{\sqrt {89} }}\]
Now putting both the above values of \[\sqrt {26} = 5.099\] and \[\sqrt {89} = 9.434\] in the place of \[\dfrac{{\sqrt {26} }}{{\sqrt {89} }}\] we get,
\[ \Rightarrow \dfrac{{\sqrt {26} }}{{\sqrt {89} }} = \dfrac{{5.099}}{{9.434}}\]
After dividing we get,
\[ \Rightarrow \dfrac{{5.099}}{{9.434}} \approx 0.54049\]
That is the required solution.
Therefore, the square root of \[\dfrac{{26}}{{89}}\] is approximately \[0.54049\].

Note:
Irrational numbers are those kinds of real numbers which cannot be written in the form of \[\dfrac{p}{q}\] where \[p\] and \[q\] are both integers and \[q \ne 0\] . Radicals of those numbers which are not a perfect square are not integers, and hence they and their fractions are irrational numbers. Ex- \[\sqrt 5 ,\dfrac{{\sqrt 7 }}{3},\dfrac{1}{{\sqrt 3 }}\] are irrational numbers.