Question

How do you find \[\sqrt {\dfrac{9}{{49}}} \]?

Answer 1

Hint: To solve this question, we first need to separate the root for the numerator and the denominator. Then, since both the numerator and the denominator are the perfect squares, the given number can be written as the ratio of two whole numbers. Then we have to perform the required division to finally express the given number in decimal form.

Complete step-by-step solution:
Let the number given in the question be equal to some variable, say \[x\]. Therefore, we can write
\[x = \sqrt {\dfrac{9}{{49}}} \]
\[ \Rightarrow x = \sqrt {9 \times \dfrac{1}{{49}}} \]
Now, we know from the laws of exponents that \[{\left( {ab} \right)^n} = {a^n}{b^n}\]. So the above equation can be written as
\[\begin{array}{l}x = \sqrt 9 \dfrac{1}{{\sqrt {49} }}\\ \Rightarrow x = \dfrac{{\sqrt 9 }}{{\sqrt {49} }}\end{array}\]
Now, we know that \[\sqrt 9 = 3\] and \[\sqrt {49} = 7\]. Substituting these in the above equation, we get
\[x = \dfrac{3}{7}\]
Now, to find the decimal form of the given number, we carry out the divide \[3\] by \[7\]. On dividing we finally get
\[x = 0.\overline {42857} \]

Hence, the value of \[\sqrt {\dfrac{9}{{49}}} \] is equal to \[0.\overline {42857} \].

Note:
We might make a mistake by carrying out the division before simplifying the square root. That is, we should not divide \[9\] by \[49\] before taking the square root. Thus, the result of the division will not be a perfect square and on taking its square root, we will get an irrational number. We know that irrational numbers are the numbers in which the digits after the decimal are non-terminating non-repeating. But, as we can see in the above solution that the given number is equal to \[0.\overline {42857} \], which is non terminating but repeating. This means that the given number is a rational number.