Question

How do we rationalize the denominator and simplify \[\dfrac{1}{{\sqrt {11} }}\]?

Answer 1

Hint: In this sum in place of the denominator, the square root of a number is given. Square root is an inverse operation of squaring a number. When the exponent is \[2\] , it is called a square whereas when the exponent is \[\dfrac{1}{2}\] , it is called a square root of a number. Square root of the number is generally multiplied by itself to get the original number.
In order to simplify this sum we need to rationalize the denominator. That is we must find some ways to convert the fraction into the form where the denominator has only rational (fractional or whole number) values. In order to rationalize the denominator we have to get rid of all the radicals that are in the denominator.

Complete step-by-step answer:
A Rational number can be defined as any number which can be represented in the form \[\dfrac{P}{Q}\] where \[Q\] is not equal to zero. Irrational numbers are the real numbers which are not rational. They cannot be expressed as the ratio of two integers.
To simplify :- \[\dfrac{1}{{\sqrt {11} }}\]
Here the numerator is \[1\] and the denominator is \[\sqrt {11} \] .
It is clear that the denominator is an irrational number. As \[\sqrt {11} \]cannot be expressed as the ratio of two integers. It cannot be represented as a simple fraction.
We need to rationalize the denominator to simplify the sum;
First we have to multiply the numerator and denominator by a radical that will help to get rid of that. So we need to multiply \[\sqrt {11} \] with numerator and denominator.
We get, \[\dfrac{{\left( {1 \times \sqrt {11} } \right)}}{{\left( {\sqrt {11} \times \sqrt {11} } \right)}} = \dfrac{{\sqrt {11} }}{{11}}\] ; this is the simplified answer.

Note: We need to keep in mind that multiplying the numerator and denominator by the exact same thing will keep the fraction equivalent. When the square root of a number is multiplied by itself we get back the original number. We must have a clear conception with rational and irrational numbers. We must know \[0\] is a rational number as it can be expressed as \[\left( {\dfrac{0}{2}} \right)\] or \[\left( {\dfrac{0}{4}} \right)\] ; it can be written in the form of a ratio.