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How to solve Surds part 3, Surd expression comparison and ranking

Sum of Surds comparison is the basis for sum of surds ranking


We have given a brief introduction to surds in our article Fractions and decimals part 1and methods on how to solve surd problems in two tutorials, How to solve surds 1, Rationalization and How to solve surds 2, Double square root surds and Surd term factoring. You may like to go through these before going ahead further in this session.

In this session we will learn about the last round of concepts and methods comprising of,

  1. Subtractive or additive two-term Surd expression comparison:  As you know surds are irrational numbers, you can see them written down on paper, but don't know their value. Individual surd term comparison is not a problem as you rank them by the numbers under the square root. We are not discussing about individual surd term comparison or ranking here at all. It is about two-term surd  expressions, where it gets difficult to assess whether one such expression $(\sqrt{7}-\sqrt{5})$ is larger or smaller than say, $(\sqrt{13}-\sqrt{11})$.
  2. Two-term surd expression ranking: If you know how to compare two numbers of two-term surd expressions, you might be able to rank a given number of such expressions as well.

These form a rather awkward group of problem classes for which you may not always be able to reach the answer quickly enough for meeting the requirement of a MCQ based competitive test without using advanced concepts and methods.

To aid the understanding of quick comparison of a pair of sum of surds expressions of specific nature, let us for state and then prove two powerful rich concepts that can be applied to quickly decide which of the expressions is larger.

Equal difference surd comparison concept

Let us state the important rich concept of surd sum comparison.

If in two subtractive two-term surd expressions, the difference between the two terms values under square root are same for the two surd expressions, then the expression with the lesser value of positive surd term will be the larger.

In other words, comparing two subtractive two-term surd expressions,

$\sqrt{a}-\sqrt{b}$, and


if $a-b=c-d$, then whichever of the positive term values $a$ or $c$ is smaller will have its associated surd expression larger.

Though it seems a bit complex to state and understand, using the concept is straightforward, especially with numeric surd terms.

To increase your belief in the validity of this apparently involved rich concept we will provide an easy to understand proof.

Proof of equal difference surd comparison concept

The subtractive surd sums are,

$\sqrt{a}-\sqrt{b}$, and


Also these have the interesting property that the difference of two term values under square root in each expression is same,


Let us assume,

$a \gt c$.

We have,


Or, $a-c=b-d$.

As $a \gt c$, $b \gt d$.

Taking up the surd expressions further,

$\sqrt{a}-\sqrt{b}=\displaystyle\frac{a-b}{\sqrt{a}+\sqrt{b}}$, multiplying and dividing by $\sqrt{a}+\sqrt{b}$.



Taking the ratio,

$\displaystyle\frac{\sqrt{a}-\sqrt{b}}{\sqrt{c}-\sqrt{d}}=\frac{\sqrt{c}+\sqrt{d}}{\sqrt{a}+\sqrt{b}}$, as $(a-b)=(c-d)$ these cancel out.

As $a \gt c$ and $b \gt d$, 

$\sqrt{a}+\sqrt{b} \gt \sqrt{c}+\sqrt{d}$.


$\displaystyle\frac{\sqrt{a}-\sqrt{b}}{\sqrt{c}-\sqrt{d}} \lt 1$,

Or, $\sqrt{a} - \sqrt{b} \lt \sqrt{c}-\sqrt{d}$.

Similarly, if $c \gt a$, the result would be reverse, that is,

$\sqrt{c}-\sqrt{d} \lt \sqrt{a}-\sqrt{b}$.

A second rich concept follows from this powerful pattern based surd comparison rich concept. We will explain it next.

Equal sum surd comparison concept

Let us state this rich surd expression comparison concept that is based on the equal difference surd comparison concept.

If in two surd expressions,

$\sqrt{a}+\sqrt{b}$ and $\sqrt{c}+\sqrt{d}$,

the sum of the terms under roots are equal,



$a \gt c$, where $a$ and $c$ are the larger of the two term values in the two expressions respectively


$\sqrt{a}+\sqrt{b} \lt \sqrt{c}+\sqrt{d}$.

Mechanism of the rich concept of Equal sum surd comparison concept

The two expressions to be compared are,

$\sqrt{a}+\sqrt{b}$ and $\sqrt{c}+\sqrt{d}$.



Or, $a-d=c-b$.

As $a \gt c$, where $a$ and $c$ are the larger terms in each expression, by equal difference surd comparison concept,

$\sqrt{a}-\sqrt{d} \lt \sqrt{c}-\sqrt{b}$,

Or, $\sqrt{a}+\sqrt{b} \lt \sqrt{c}+\sqrt{d}$, by inequality arithmetic.

This second rich pattern based concept follows from the first.

Through solving of chosen SSC CGL level problems we will show how to apply these two powerful methods.

Problem example 1.

Which of the following is the correct relation?

  1. $\sqrt{5}+\sqrt{3} \lt \sqrt{6}+\sqrt{2}$
  2. $\sqrt{5}+\sqrt{3} = \sqrt{6}+\sqrt{2}$
  3. $\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$
  4. $\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{6}+\sqrt{2}\right)=1$

Solution 1: Problem analysis

In surd expression comparison, one of the most important concepts that we use frequently is what we call, the Equal difference surd comparison concept.

In this problem we will apply the concept if we can.

Solution 1: Problem solving execution

In all four options the same two surd expressions appear. Let us evaluate the comparative relation between these two expressions.

By equal difference surd comparison concept,

$\sqrt{6}-\sqrt{5} \lt \sqrt{3}-\sqrt{2}$,

Or, $\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$.

So only Option: $c$ is true.

Answer: Option c: $\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$.

Key concepts used: inequality concepts -- equal difference surd comparison concept.

Faster solution 1

The sum of terms under square root of both expressions is 8. So by equal sum surd comparison concept,

$\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$.

Problem Example 2.

Which is the greatest among $(\sqrt{19}-\sqrt{17})$, $(\sqrt{13}-\sqrt{11})$, $(\sqrt{7}-\sqrt{5})$, and $(\sqrt{5}-\sqrt{3})$?

  1. $(\sqrt{5}-\sqrt{3})$
  2. $(\sqrt{7}-\sqrt{5})$
  3. $(\sqrt{19}-\sqrt{17})$
  4. $(\sqrt{13}-\sqrt{11})$

Solution 2: Problem analysis and solving execution

Examining all the four terms we find the difference between two values under square roots for each expression is equal to 2. Among all four expressions, $(\sqrt{5}-\sqrt{3})$ having the smallest value of the positive term under square roots $5$, this expression will be the largest by the equal difference surd comparison concept.

We need just to identify the key pattern and apply the powerful equal difference surd comparison concept.

Answer: Option a: $(\sqrt{5}-\sqrt{3})$.

Key concepts used: Key pattern identification -- Equal difference surd comparison.

Problem example 3.

The smallest of $\sqrt{8}+\sqrt{5}$, $\sqrt{7}+\sqrt{6}$, $\sqrt{10}+\sqrt{3}$, $\sqrt{11}+\sqrt{2}$ is,

  1. $\sqrt{7}+\sqrt{6}$
  2. $\sqrt{8}+\sqrt{5}$
  3. $\sqrt{10}+\sqrt{3}$
  4. $\sqrt{11}+\sqrt{2}$

Solution 3: Problem analysis and solving

Examining the four surd expressions we find that the expressions are neither in sutractive form nor the term differences are equal. But still we find possibility of comparing the first two expressions and the last two expressions by transposing terms suitably.

Let us then compare the first two expressions,

$\sqrt{8}+\sqrt{5}$, $\sqrt{7}+\sqrt{6}$.

Observing that if we subtract $\sqrt{7}$ from $\sqrt{8}$ and again subtract $\sqrt{5}$ from $\sqrt{6}$ we will have a conclusive relation by the equal difference surd comparison concept,

$\sqrt{8}-\sqrt{7} \lt \sqrt{6}-\sqrt{5}$,

Or, $\sqrt{8}+\sqrt{5} \lt \sqrt{7}+\sqrt{6}$, by basic inequality arithmetic.

Similarly we will have,

$\sqrt{10}+\sqrt{3}\gt \sqrt{11}+\sqrt{2}$.

Now we have to compare the two smaller valued expressions,

$\sqrt{8}+\sqrt{5}$ and $\sqrt{11}+\sqrt{2}$.

Again we find the possibility of applying the equal difference surd comparison concept by transposing terms,

$\sqrt{11}-\sqrt{8} \lt \sqrt{5}-\sqrt{2}$,

Or, $\sqrt{11}+\sqrt{2} \lt \sqrt{8}+\sqrt{5}$.

Thus, $\sqrt{11}+\sqrt{2}$ will be the smallest among the four.

Answer: Option d: $\sqrt{11}+\sqrt{2}$.

Key concepts used: Key pattern identification -- Inequality analysis by transposition -- Inequality arithmetic -- Equal difference surd comparison concept.

Quicker Alternate solution: By applying equal sum surd expression comparison concept

Sum of the surd terms under roots for each of the four given expressions is equal to 13. The larger values in the expressions are, 7, 8, 10 and 11. So the expression with the largest of the larger values, $\sqrt{11}+\sqrt{2}$ will be the smallest.

This is by far the fastest solution because we have applied the second level rich surd expression comparison concept derived from first level rich concept.

Guided help on Fractions, Surds and Indices in Suresolv

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