In Mathematics, the word rationalisation is used when we need to remove a radical term from the denominator. A denominator is basically the lower part of a fraction. A fraction is a part of a whole. For example, ¼ is a fraction, where 1 is the numerator, and 4 is the denominator.
Some fractions are simple to solve and express in a standard form. Let us say, we need to write 4/8 into standard form. Since the common factor of 4 and 8 is 4, therefore, we can write 4/8 equal to (4×1)/(4×2) = ½. Thus, with the concept of factors, we can write the fraction in the simplest form. But what if the denominator has a radical or surds in it. In this article, we will learn the rationalisation of such denominators containing radical terms.
How to Rationalise the Denominator?
In a fraction, if we have a radical term or a surd, we need to get rid of it to express it in a standard form. A radical term is expressed using the symbol ‘√’. It is also called square root. An example of a fraction with surd is 2/√3. To simplify 2/√3, we need to multiply and divide the given fraction by √3.
2/√3 x (√3/√3) = 2√3/3
The denominator in the above example is a monomial. But it is not necessary that a denominator is always a monomial. It could be a linear equation or a binomial too. This procedure might be reached out to any mathematical denominator, by increasing the numerator and the denominator by all arithmetical forms of the denominator and growing the new denominator into the standard of the old denominator. In any case, the subsequent fractions might have tremendous numerators and denominators, and in this manner, the strategy is, for the most part, utilised uniquely in the above rudimentary cases. Now, if we are asked to simplify such expressions, then we need to rationalize the denominator. Let us solve some examples to understand.
Example 1: Simplify the expression: 10/√5
To rationalise the expression, we need to multiply and divide by √5
10/√5 x (√5/√5)
= 10√5/(√5)2
= 10√5/5
= 2√5
What if the radical is not a square root but a cube root? Let us see how we can simplify such rational numbers.
Example 2: Simplify the following: 2/3√3.
Since the denominator is a cube root of 3, therefore, we need to multiply both numerator and denominator by the factor (3√3)2, to make the denominator a whole number.
2/3√3 x (3√3/3√3)2
= 23√32/( 3√3x3√32)
= 23√3/(3√3)3
= 23√3/3
More of Square Roots
If the denominator is of the form of a+√b, then we have to multiply both numerator and denominator by a-√b, to rationalise the denominator.
Example 3: Rationalise the denominator and write the expression in the simplest form.
2/(4 + √7)
The denominator here is 4 + √7. It means there are two terms. Thus, we can just multiply and divide by radical terms to remove the surds from the denominator.
So multiplying and dividing 2/4+√7 by 4-√7 on numerator and denominator, we get;
⇒ (2/(4+√7)) x (4-√7)/(4-√7)
Here, we will use the algebraic identity;
a2 – b2 = (a-b) (a+b)
Therefore,
⇒ 2(4-√7)/(42 – √72)
⇒ 2(4-√7)/(16 – 7)
⇒ 2(4-√7)/9
Hence, this is the simplest form of the given expression.
By this article, we have now received the knowledge of rationalisation in Mathematics and how to express a fraction into simplest form.