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How to Solve Surds Part 1 - Rationalization of Surds

Rationalization of Surds - How to Solve Surds Part 1

Rationalizaion of surds - How it works and how to use the Surds problem solving technique

Rationalization of Surds is explained through example problems and actual test level problems. This is the first part in the series on How to Solve Surds.

By rationalization of surds you can convert, for example, the surd expression $(5-\sqrt{3})$ to the rational integer 22. This simplifies surds problem solving.

In this first part of the series on How to solve surds, you will learn,

  1. What are surds in brief,
  2. What is rationalization of surds,
  3. How rationalization of surds works,
  4. How to solve surds problems by rationalization of surds: How to solve surd problems by rationalization of surds explained through solving actual test level problems.

To skip the next section and go straight to What is Rationalization of surds, click here.

What are surds

Surds are a group of irrational numbers expressed as,

$\sqrt{\text{Product of Prime number factors}}$.

Square root of any prime number like 3 is a surd $\sqrt{3}$.

Examples of surds are,

$\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{12}$ and so on.

In a surd, the number under square root must have at least one unique prime factor not equal to any of its other factors.

For example, in $\sqrt{12}$, 12 has three factors 2, 2 and 3. Only, prime number 3 is unique and not equal to the other two factors of 2.

In $\sqrt{15}$, two unequal factors are both prime.

At first, surds problems may look difficult to solve. But solving surds is pretty easy if you know how to apply only a few surds problem solving techniques.

Among these, rationalization of surds is the simplest.

Multiplication, Division and Indices on surds

An example of surd multiplication is,


The multiplication happens within the square root resulting in the product 21, but the result remains a surd.

An example of surd division is,

$\displaystyle\frac{\sqrt{2}}{\sqrt{5}}=\sqrt{0.4}$, which is again a surd ($\sqrt{0.4} \neq 0.2$, square of 0.2 is 0.04).

Though after the operation number of terms reduces, the result still remains a surd.

Indices on surds is just an extension of existing indices concepts because a surd term has a power of $\displaystyle\frac{1}{2}$,

$\left(\sqrt{3}\right)^3=3^{\frac{3}{2}}$, still a surd.

To skip the next section and go straight to How rationalization of surds works click here.

What is rationalization of surds

In simple words,

Rationalization of surds converts an irrational surd expression to a rational number (or expression in rational variables).

For example, by rationalization of surds you can convert the surd expression $(5-2\sqrt{3})$ to the rational number 13.

In general, rationalization of surds,

Converts a two-term IRRATIONAL SURD EXPRESSION of the form $\sqrt{a}\pm\sqrt{b}$ to a RATIONAL EXPRESSION $(a-b)$, where both $a$ and $b$ are rational.

Other than conversion from irrational to rational, observe two specific properties of the operation of rationalization of surds,

1. For applying rationalization of surds on a surd expression, it must be a two-term surd of the form $(\sqrt{a} \pm \sqrt{b})$.

    For example, $2-\sqrt{3}$ or $\sqrt{19}+3\sqrt{2}$ can be rationalized, but $2+\sqrt{3}+\sqrt{5}$ cannot be.

2. The result of rationalization of surds is fixed and known. For example, if you rationalize the surd expression $7+\sqrt{13}$ the result will be $7^2-13=36$.

Let us see how rationalization of surds works.

How rationalization of surds works

Rationalization of surds is based on the algebraic identity,


When either one or both of $a$ and $b$ are standalone single term surds, because of the squaring effect, result become rational.

As an example let us rationalize the DENOMINATOR OF THE SURD EXPRESSION,


To rationalize the denominator $(2-\sqrt{3})$, multiply and divide the surd fraction by the surd expression $(2+\sqrt{3})$ which is the expression complementary to $(2-\sqrt{3})$.

Result of the operation will be,


As the denominator turns to a number, the surd denominator is effectively eliminated and the target expression is simplified.

Let us solve a few actual test level problems to get a feel of how surd problems are solved by rationalization of surds.

How to solve surds problems by rationalization of surds

Problem example 1.

What is the simplified value of $\displaystyle\frac{1}{\sqrt{2}+1}+3-\sqrt{2}$?

  1. $1$
  2. $0$
  3. $2$
  4. $3$

Solution to Problem example 1.

Simplification by normal means seems to be difficult. In this general situation of simplifying a sum of fraction terms try to,

Eliminate the surd in the denominator.

This is the general problem solving strategy of denominator elimination.

Rationalization of surds is usually done for eliminating surd denominator, but occasionally rationalization of surds may be neeed to be applied on numerator surd also.

To eliminate the surd in the denominator, we'll use the concept inherent in the algebraic expression,

$(a+b)(a-b)=a^2-b^2$, where both $a$ and $b$ may be surds.

As the individual terms gets squared, surds terms are eliminated from the result.

Applying the concept, the fraction term is multiplied and divided by the surd expression complementary to the denominator, in this case by $\sqrt{2}-1$.

Note: While deciding on this multiplying factor, care is to be taken to keep the larger term positive. For example don't use $1 - \sqrt{2}$, but use $\sqrt{2}-1$ as $\sqrt{2} \gt 1$ and it will ensure a positive result.

Result of the action is,





Knowing that the surd fraction can be replaced by its complementary surd expression, we do this automatically in mind.

That's the advantage of the technique of rationalization surds. It can be done always in mind.

Solution to Problem example 1.

Rationalize the denominator of the first term,




Answer: Option c: 2.

A simple and quick result.

Problem example 2.

Value of $\displaystyle\frac{1}{3-\sqrt{8}}-\displaystyle\frac{1}{\sqrt{8}-\sqrt{7}}+\displaystyle\frac{1}{\sqrt{7}-\sqrt{6}}$

$\hspace{30mm}-\displaystyle\frac{1}{\sqrt{6}-\sqrt{5}}+\displaystyle\frac{1}{\sqrt{5}-2}$ is,

  1. 2
  2. 3
  3. 4
  4. 5

Solution 2. Problem example 2.

Each of the denominators has one unique property,

When squared, the difference between the two terms becomes 1.

This helps.

After a denominator, say, $\sqrt{p}-\sqrt{q}$, is rationalized by multiplying both numerator and denominator by its complementary surd, $\sqrt{p}+\sqrt{q}$, the denominator is transformed to,

$\left(\sqrt{p}\right)^2 - \left(\sqrt{q}\right)^2=p-q=1$.

So after rationalization, each denominator will be eliminated.

Note: For convenience of mental processing we will always assume $3=\sqrt{9}$ and $2=\sqrt{4}$—a uniform look aids mental processing.

What would be the numerators?

Let us jot down the expression of the sum of numerators,

$(\sqrt{9}+\sqrt{8}) - (\sqrt{8}+\sqrt{7})+(\sqrt{7}+\sqrt{6})$


All terms except the first and the last are canceled out.

So final result is,


Answer: Option d: 5.

Once you visualize the patterns, solution comes immediately.

Problem example 3.

The simplified value of $\displaystyle\frac{3\sqrt{2}}{\sqrt{3}+\sqrt{6}}-\displaystyle\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}}+\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$ is,

  1. $\sqrt{2}$
  2. $\sqrt{3}-\sqrt{2}$
  3. $0$
  4. $\displaystyle\frac{1}{\sqrt{2}}$

Solution to problem example 3.

Rationalization of surds must be done on all the denominators. But before taking this inevitable step,

First try to simplify the target terms by themselves.

This is the golden strategy of Target expression simplification first.

On closer look, identify the factor of $\sqrt{3}$ COMMON between the two terms of the first denominator and the factor of $\sqrt{2}$ between the two terms of the second denominator.

Take these factors out of the two terms in both denominators.

The target expression becomes,



Identify also that $\sqrt{3}$ in the first term and $\sqrt{2}$ in the second could be cancelled out with the numerator.

After this, each numerator gets the same factor of $\sqrt{6}$.

As the three denominators are simplified fully, apply rationalization of surds on all the three.

Result of the action is,


$=0$, as all the terms under the brackets cancel out.

You could very well have solved the problem all in mind.

Answer: Option c: $0$.

So you know by now how to solve actual test level surds problems.

You may wonder - why only the denominator is rationalized in all the problems? Is it true that rationalization of surds can be done only on denominators?

Answer is a firm NO.

Rationalization of surds is a mathematical concept as well as a technique. It depends on you how you would use it.

We'll show you a more difficult problem example where rationalization of surds is done on the numerator.

Problem example 4: Rationalization of surds numerator

What is the value relationship between the irrational numbers, $\sqrt{7} - \sqrt{5}$, $\sqrt{5} - \sqrt{3}$ and $3 - \sqrt{7}$?

  1. $3 - \sqrt{7} \lt \sqrt{5} - \sqrt{3} \lt \sqrt{7} - \sqrt{5}$
  2. $\sqrt{5} - \sqrt{3} \lt 3 - \sqrt{7} \lt \sqrt{7} - \sqrt{5}$
  3. $3 - \sqrt{7} \lt \sqrt{7} - \sqrt{5} \lt \sqrt{5} - \sqrt{3}$
  4. $3 - \sqrt{7} \gt \sqrt{7} - \sqrt{5} \gt \sqrt{5} - \sqrt{3}$

Solution to problem example 4:

In this subtraction form of surd expressions, you can't be mathematically certain about which one of the given surd expressions is larger or smaller than another.

Searching for some pattern in the three expressions, you detect one minor pattern, especially when you transform the third given expression and place it in the beginning as,

$\sqrt{9} - \sqrt{7}$, $\sqrt{7} - \sqrt{5}$, and $\sqrt{5} - \sqrt{3}$.

$\sqrt{7}$ is common between the first and second expressions and $\sqrt{5}$ is common between the second and third expressions.

But unfortunately these are in opposite signs.

At this point you realize, if you could have converted these surd expressions with the same terms but in addition form, evaluation of relative values would have been straightforward.

And in fact, this conversion of a two-term subtractive surd sum to additive surd sum is easily possible by Rationalization of surds.

Here it works perfectly as you identify the second key pattern in the six surd terms of the three expressions,

Difference between the two numbers under square root for each expression is 2.

Rationalize the three given expressions by multiplying and dividing the first, the second and the third by $(\sqrt{9}+\sqrt{7})$, $(\sqrt{7}+\sqrt{5})$, and, $(\sqrt{5}+\sqrt{3})$ respectively.

The results you would get are,

$\displaystyle\frac{9-7}{\sqrt{9} + \sqrt{7}} = \displaystyle\frac{2}{\sqrt{9} + \sqrt{7}}$,

$\displaystyle\frac{7-5}{\sqrt{7} + \sqrt{5}} = \displaystyle\frac{2}{\sqrt{7} + \sqrt{5}}$, and,

$\displaystyle\frac{5-3}{\sqrt{5} + \sqrt{3}} = \displaystyle\frac{2}{\sqrt{5} + \sqrt{3}}$.

Which one will be the largest?

By basic division and fraction concepts,

Among two fractions with equal numerators, the fraction with smaller denominator will be the larger of the two.

In our results, the numerators are all equal at 2.

So the fraction that has the largest denominator will be the lowest and the corresponding original surd sum will also be the lowest.

$\sqrt{9}+\sqrt{7}$ is the largest denomnator. So its corresponding original surd sum $3-\sqrt{7}$ is the smallest of the three.

Next smaller denominator $\sqrt{7}+\sqrt{5}$ will identify the middle value of $\sqrt{7}-\sqrt{5}$.

Smallest denominator of the three $\sqrt{5}+\sqrt{3}$ will correspond to largest given surd sum $\sqrt{5}-\sqrt{3}$.

The relative values of the given surds expressions would then be,

$(3 - \sqrt{7}) \lt (\sqrt{7} - \sqrt{5}) \lt (\sqrt{5} - \sqrt{3})$.

Answer: Option c: $3 - \sqrt{7} \lt \sqrt{7} - \sqrt{5} \lt \sqrt{5} - \sqrt{3}$.

You may wonder, is it possible to solve this awkward problem in mind!

Knowing how to apply the rich concept of ranking subtractive two term surd expressions, it should take just a few tens of seconds to arrive at the answer.

In these three two-term surd expressions we would identify the patterns,

Two equal adjoining common terms between first-second and second-third terms, and,

Equal difference of the two numbers under square roots for each term.

These two patterns will decide $3-\sqrt{7}$ as the smallest and $\sqrt{5}-\sqrt{3}$ as the largest.

End note

By itself, rationalization of surds is a simple concept and technique. But the problems it helps to solve may be hard to solve.

Next read

To know all the surd problem solving techniques, go through our concept articles,

How to solve surds part 2, double square toot surds and surd term factoring.

How to solve surds part 3, surd comparison and ranking.

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