Rationalization is a method that is used to eliminate irrational numbers. The meaning of rationalizing is making something more efficient. Irrational numbers are those which can’t be expressed in a simple fraction. Check the rules and methods to rationalize an irrational number and its definition in the following sections. Also, know the example questions along with answers.
What is Meant by Rationalization?
rationalization is a process that is used to eliminate the irrational number from the denominator of an algebraic function. It removes the radicals in a fraction thus the denominator has only one rational number. The important terms of rationalization are here.
- Radical: It is an expression that uses roots like square root, cube root.
- Radicand: It is a term that finds the root of. For example, √(x + y) the term (x + y is the radicand.
- Degree: It is the number present at the root symbol. 2 is square root, 3 is cube root.
- Radical Symbol: The radical symbol √ means the root of. The length of the horizontal bar signifies the part of the root function.
- Conjugate: The conjugate of any binomial means another binomial with an exact opposite sign between its two terms.
Do Check:
- Decimal Representation of Irrational Number
- Representation of Irrational Numbers on The Number Line
- Comparison between Rational and Irrational Numbers
Rationalization of Monomial Radical
To rationalize a radical which in the denominator of a fraction, you have to follow these steps.
- If the denominator has a radical in the form of \(\frac { a }{ √b } \). Then the fraction must be multiplied and divided by √b.
- For a polynomial with monomial radical in the denominator in the form of an nth root of xn, such that n < m, then the fraction must be multiplied by a quotient having an nth root of xm – n in both numerator and denominator.
Rationalization by Multiplication with Conjugate
In the fractions that have irrational numbers in the form of addition, subtraction in the denominators of the fractions, we use this multiplication with a conjugate method for rationalizing the fraction.
We have an expression (x + √y)(x – √y) = x² – (√y)² = x² – y which is a rational number.
Therefore, multiplying an irrational number by another irrational number gives the rational number. Here, (x – √y) is a rationalising factor of (x + √y). In the same way (x + √y) is a rationalising factor of (x – √y). The irrational number (x – √y) is called the conjugate irrational number or conjugate of (x + √y). Similarly, (x + √y) is the conjugate of (x – √y).
Problems on Rationalizing Irrational Numbers
Problem 1:
Rationalize the fraction \(\frac { 2 }{ √3 } \)
Solution:
Given fraction is \(\frac { 2 }{ √3 } \)
Since √3 is an irrational number and it is there in the denominator of the fraction. So, we need to rationalize it. This can be done by multiplying both numerator and denominator with √3.
\(\frac { 2 }{ √3 } \) = \(\frac { 2 }{ √3 } \) x \(\frac { √3 }{ √3 } \)
= \(\frac { 2√3 }{ 3 } \)
Problem 2:
Rationalize \(\frac { 1 }{ 5 + √2 } \).
Solution:
Given fraction is \(\frac { 1 }{ 5 + √2 } \)
Since, the given problem has an irrational term in the denominator with addition and subtraction format. So we need to rationalize using the method of multiplication by the conjugate. So,
\(\frac { 1 }{ 5 + √2 } \) = \(\frac { 1 }{ 5 + √2 } \) x \(\frac { 5 – √2 }{ 5 – √2 } \)
= \(\frac { 5 – √2 }{ 5² – √2² } \)
= \(\frac { 5 – √2 }{ 25 – 2 } \) = \(\frac { 5 – √2 }{ 23 } \)
So, the rationalized number is \(\frac { 5 – √2 }{ 23 } \).
Problem 3:
Rationalize \(\frac { 1 }{ √2 + √5 } \).
Solution:
Given fraction is \(\frac { 1 }{ √2 + √5 } \)
Since, the given problem has an irrational term in the denominator with addition and subtraction format. So we need to rationalize using the method of multiplication by the conjugate. So,
\(\frac { 1 }{ √2 + √5 } \) = \(\frac { 1 }{ √2 + √5 } \) x \(\frac { √2 – √5 }{ √2 – √5 } \)
= \(\frac { √2 – √5 }{ √2² – √5² } \)
= \(\frac { √2 – √5 }{ 2 – 5 } \)
= \(\frac { √2 – √5 }{ -3 } \)
= \(\frac { √5 – √2 }{ 3 } \)
Therefore, the rationalized number is \(\frac { √5 – √2 }{ 3 } \).
FAQ’s on Factorization
1. Why do we use rationalization?
We use rationalization to rationalize the denominator of the fraction which is having an irrational number so that it is very easy to perform calculation on the obtained rational number.
2. What is an example of rationalization?
A fraction with irrational number in the denominator can be rationalized by removing radical. For example in the fraction \(\frac { 1 }{ √2 } \). Multiply numerator, denominator by √2 i.e \(\frac { √2 }{ 2 } \).
3. What are the rules of rationalization?
The simple rules of rationalization are here.
- Check for the radicals in the simplified form.
- Find the suitable radical and multiply it in the numerator and denominator that will eliminate radicals in the denominator.