Rationalizing the Denominator

Previously we have discussed about area of octagon calculator and In today's session we are going to discuss about Rationalizing the Denominator which comes under andhra pradesh state education board, It defines the method to rewrite a fraction in a form containing merely rational numbers in the denominator. Finding the equivalent expression (which in the denominator not have any radicals) of a radical expression is required in algebra and is known as rationalizing the denominator. In simple words when moving a root value from the bottom of a fraction to the top of fraction is called as rationalizing the denominator . There are basically three types of cases in algebra( check out Algebra Answers for reference) that are:
( a ) square root
( b ) cube root or other roots of higher category
( c ) square root addition or difference
In case ( a ), when in the denominator there is a only single square root then the rationalization process of denominator is done only by multiplying numerator and denominator with the square root. We can understand it by an example as 1 / √ 3 is rationalized by multiplying √ 3 to both numerator and denominator of the fraction. (Know more about  in broad manner, here,)
( 1 / √ 3 ) * √ 3 / √ 3 = √ 3 / 3
In case ( b ) when there is higher root (means cube root etc) in the denominator then we multiply and divide the fraction with something that provides the perfect power. For example if there is cube root of 3 in the denominator than we can multiply it by 9 that is the 2nd power of 3 .
We can understand it also by an example as 1 / ³√ p q that is solved as
1 / ³√ p q = 1 / √ p q * ³√ p ² q ² / ³√ p ³ q ³ = ³√ p² q ²/ p q
In case ( c ) numerator and denominator are multiplied with their difference which can be sum or vice versa . To understand it, one example is discussed that is 7 / 9 + √ 10 then
= 7 / 9 + √ 10 * 9 - √ 10 / 9 - √ 10
= 7 ( 9 - √ 10 ) / 81 – 10
= 63 – 7 √ 10 / 71 .
In the next session we are going to discuss about fractional notation calculator
and if anyone want to know about Inequalities then they can refer to Internet and text books for understanding it more precisely.

Rationalizing the Denominator

Previously we have discussed about adding scientific notation calculator and In today's session we are going to discuss about Rationalizing the Denominator which comes under andhra pradesh state board of secondary education,  The process of eliminating a square root or imaginary number from the denominator of a fraction is termed as rationalization. A rational number is a number that can be expressed as the ratio of two integers, such as 2/3. Any number with a terminating decimal part is a rational number. If decimal part begins to repeat in a numeral, it is also rational number, such as .0303030..., since this can be expressed as 1/33.  Numbers that are not rational are known as irrational.  Examples of irrational numbers are the square root of 2, pi, and e. Examples of rational no are 2/4,3/5.

Rationalizing the Denominator is done to simplifying fractions into simplest form i.e. the denominator should not have an irrational number or a complex number.
In case of real number there are 3 cases where we use rationalization
1. The denominator having single square root
2. The denominator having single higher root
3. The denominator having sums and differences of square roots
case 1:
 When you have a single square root in the denominator you just multiply top and bottom by it.
Example:

1/
=1*/*

=1*/3
in th above example  the denominator is . so we need to rartionaze it forthis we multiply and divide it by the same no . doing this the value of the expression remains same and the square root gets removed from the denominator as * gives 3 which is not irrational

case 2:When we have denominators having higher root
example:   1/

=1/(*)

 = 1/a
since this is a 3rd root, in order to remove the root from the denominator we have to get cube of the values inside the root. To get this we multiply the denominator and numerator with the cube root of the square of the value. Finally we get the cube root of the cube of the value and thus the cube root is removed and we get a rational no in the denominator. (Know more about Rationalizing the Denominator in broad manner, here,)
CASE 3: When the denominator is having   sums and differences of square roots
example:2/1-

=2*(1+)/(1-)(1+)

=2*(1+)/1-3

=2*(1+)/-2

=-(1+)

In this case if a sum is in the denominator we  multiply the denominator and numerator with the difference. And if we have a difference in the denominator we multiply the denominator and numerator with the sum. Thus we rationalize denominator.
In The Next Session We Are Going to Discuss Equations with no Solution

and Read more maths topics of different grades such as Equations and Inequalities in the upcoming sessions here.