Question

How do you rationalize the denominator and simplify\[\dfrac{{\sqrt {22} }}{{\sqrt {55} }}\]?

Answer 1

Hint: The given question need to be simplified by rationalizing the denominator for which we need to multiply and divide the factor given in the denominator into the fraction given, and then the root of the denominator will get remove and then we can simplify further to get the simplified fraction.

Complete step by step solution:
The above fraction need to be rationalize and then solve further, to rationalize the denominator we need to multiply and divide the denominator term with the fraction, on solving we get:
\[
   \Rightarrow \dfrac{{\sqrt {22} }}{{\sqrt {55} }} \times \dfrac{{\sqrt {55} }}{{\sqrt {55} }} = \dfrac{{\sqrt {22 \times 55} }}{{55}} = \dfrac{{\sqrt {11 \times 2 \times 11 \times 5} }}{{55}} = \dfrac{{11\sqrt {10} }}{{55}} \\
   \Rightarrow \dfrac{{11\sqrt {10} }}{{55}} = \dfrac{{11\sqrt {10} }}{{11 \times 5}} = \dfrac{{\sqrt {10} }}{5} \\
 \]
Here we obtain the fraction in which root is not available in the fraction, here we can solve further to get the answer in the decimal form by solving the numerator and denominator in the decimal form and then get the final possible least answer, but here the question ask to only rationalize the fraction hence we do not have to solve further.

Note: Here we have multiplied and divided the denominator term in the given fraction so as to get the rationalize term after solving the fraction. Rationalizing a fraction means multiplying a new fraction in the old one in which the numerator and denominator have the same term.
In a rationalize type question, we have to multiply and divide with the term given in the part of the fraction which needs to be rationalized, and after multiplying with the term we can get the square term and thus can solve further.