# Surds and Indices II- Rationalization in Surds and Indices Problems for SSC Exams

### Surds and indices problems are asked in competitive exams like SSC CGL. In the 2nd blog in this series we’ll use rationalization to solve questions on surds and indices in a few seconds!

Problems from surds and indices look really complex and appear to be very time taking, which is why a lot of you may end up skipping then in exams like SSC CGL. But with an understanding of the laws of surds and indices, these problems become really simple. In the first blog in this series, we discussed the set of laws that make surds and indices problems really simple and simplify the calculations involved. You quickly revise these laws before we move ahead to solving surds and indices problems.

### Rationalization in Surds and Indices Problems

Rationalization is the process of eliminating surds from the denominator of a fraction in surds and indices problems. Rationalization can also be used to eliminate imaginary number form a complex number. The idea behind using rationalization is to simplify the term and make calculations easier.

Let’s see how surds and indices problems are made easier by using rationalization. Take a number-
3/2

Now to solve this fraction, we need to eliminate the root term in the denominator, so how do we approach such surds and indices problems? We use rationalization. We multiply both the numerator and the denominator by a common term, to eliminate the surd in the denominator.

Example 1: For the above fraction, we know that a square root can be solved by squaring it. If 2 is multiplied by 2 it will become-
2 x 2 = (2)2 = 2
But since this is a fraction, both the numerator and the denominator have to be multiplied by the same number, so that it does not affect the fraction.
Similarly, any root in the denominator can be solved in surds and indices problems using rationalization.

Example 2: Let us now look at surds and indices problems where there is a root of a higher degree, like
Now we need to eliminate the cube root, we should have a cube. So what should we do here? We will have to multiply with 42, to get 43 but with cube root of 42.
So, this is how we multiply the numerator and denominator of a fraction with a rationalizing factor to solve surds and indices problems.

Example 3: The surds and indices problems we have solved till now have only one surd in the denominator; let’s now move to more complicated fractions that have more than one surd in the denominator. Such surd and indices problems are asked in exams like SSC CGL.
In the above fraction we have a combination of surds in the denominator. Well, the method remains the same- multiply the numerator and the denominator with a rationalizing factor. Now the next question here is, how do we get the rationalizing factor of ‘√5 + √3’

The rationalizing term of a term like this is nothing but the conjugate of the term.

The conjugate is obtained by simply negating the second term of the binomial.
So the conjugate for √5 + √3’ will be-
Multiplying both, the numerator and the denominator, with the rationalizing term we get-
Using the above algebraic expansion we get-

This is how taking the conjugate help such solve such surds and indices problems. Similar method is also used for eliminating imaginary number sin complex numbers. There we use conjugate of the imaginary number.

Example 4: Let’s look at another surds and indices problem on the same lines.
Now the approach to such surds and indices problems is really simple- multiply the numerator and the denominator with the conjugate.

So this is how we solve such problems.

### Rationalizing Factors for Surds and Indices Problems

Now let us quickly summarize what is a rationalizing factor and what are the different rationalizing factors used in surds and indices problems.