Surds

Numbers that are irrational and cannot be represented in the form of fractions or as recurring decimal numbers are known as surds. These numbers cannot be represented as recurring decimals or as fractions, they can be only represented as square roots. In other words, surds are nothing but square root representation of irrational numbers that cannot be expressed in fractional or recurring decimals. In order to make precise calculations, surds are used. In this article, you will be learning about the definition of surds, the basic six rules of the surds, some solved example problems, and the frequently asked questions related to surds.

 

Surds are the representation number in the form of square root since these numbers cannot be a whole number or a rational number. These numbers cannot be represented as fractions too. For example, consider the value of √2. The value of  √2 is said to be recurring. We know the value of  √2 = 1.414213. . . . . . . . . . .but to be accurate we will leave it as a surd. This proves that √2 is a surd.


Different Types of Surds

There are six different types of surds, namely: Simple surds, Pure Surds, Similar Surds, Mixed Surds, Compound Surds, and Binomial Surds. Now let’s understand these different types of surds.

 

  • Simple Surd: When there is only a number present in the root symbol, then it is known as a simple surd. For example √2  or √5.

  • Pure Surd: Surds that are irrational are called as pure surds. For example √3

  • Similar Surd: When surds have the same common factors, they are known as similar surds.

  • Mixed Surds: When numbers can be expressed as a product of rational and irrational numbers, it is known as a mixed surd.

  • Compound Surds: The addition or subtraction of two or more surds is known as a complex surd.

  • Binomial Surd: when two surds give rise to one single surd, the resultant surd is known as binomial surds.


Six Rules of Surds 

Rule 1:  = \[\sqrt{r X s}\] = \[\sqrt r\] x \[\sqrt s\]

Example:

Simplifying √20.

We know that, √20 = 2 x 2 x 5

    √20 = 22 x 5

Here, 4 is the greatest perfect square factor of 20

Therefore, 

\[\sqrt 20\] = \[\sqrt {2^{2}}\] x \[\sqrt 5\]

\[\sqrt 20\] = 2 x \[\sqrt 5\]

\[\sqrt 20\] = 2\[\sqrt 5\]


Rule 2: \[\sqrt{\frac{r}{s}}\] = \[\frac{\sqrt{r}}{\sqrt{s}}\]

Example: 

\[\sqrt{\frac{18}{169}}\] = \[\frac{\sqrt{18}}{\sqrt{169}}\]

\[\sqrt{\frac{18}{169}}\] = \[\frac{\sqrt{2\times3\times3}}{\sqrt{13\times3}}\]

\[\sqrt{\frac{18}{169}}\] = \[\frac{\sqrt{2\times3^{2}}}{\sqrt{13^{2}}}\]

Here, 9 and 169 are perfect squares. 

Therefore, \[\sqrt{\frac{18}{169}}\] = \[\frac{3\sqrt{2}}{13}\]


Rule 3: \[\frac{r}{\sqrt{s}}\] = \[\frac{r}{\sqrt{s}}\] X \[\frac{\sqrt{s}}{\sqrt{s}}\]

\[\frac{r}{\sqrt{s}}\] = r \[\frac{\sqrt{s}}{s}\]

The denomination can be rationalized by multiplying the numerator as well as the denominator with the value of the denominator.

Example:

\[\frac{2}{\sqrt{3}}\] = \[\frac{2}{\sqrt{3}}\] X \[\frac{\sqrt{3}}{\sqrt{3}}\]

= 2 \[\frac{\sqrt{3}}{3}\]


Rule 4: p \[\sqrt r\] ± q \[\sqrt r\] = (p ± q)\[\sqrt r\]

Example:

8 \[\sqrt 6\] ± 2 \[\sqrt 6\]

= \[\sqrt 6\](8 +2)

= 10\[\sqrt 6\]


Rule 5: \[\frac{r}{p+q^{\sqrt{n}}}\]

This rule basically helps you rationalize the denominator. You’ll have to multiply p - q\[\sqrt n\]with the numerator as well as the denominator. 

Example: 

Rationalizing 

\[\frac{1}{3+{\sqrt{2}}}\] = \[\frac{1}{3+{\sqrt{2}}}\] X \[\frac{3-{\sqrt{2}}}{3-{\sqrt{2}}}\] 

  =\[\frac{3-{\sqrt{2}}}{9-(2)}\]

  = \[\frac{3-{\sqrt{2}}}{7}\]


Rule 6: \[\frac{r}{p-q^{\sqrt{n}}}\]

This rule basically helps you rationalize the denominator. You’ll have to multiply p + q\[\sqrt n\]with the numerator as well as the denominator. 

Example: 

Rationalizing 

\[\frac{1}{3-{\sqrt{2}}}\] = \[\frac{1}{3-{\sqrt{2}}}\] X \[\frac{3+{\sqrt{2}}}{3+{\sqrt{2}}}\] 

  =\[\frac{3+{\sqrt{2}}}{9-(2)}\]

  = \[\frac{3+{\sqrt{2}}}{7}\]


Surds Formula

To solve surds easily, apply the rules that are mentioned below:


  • Rule 1:  \[\sqrt{r X s}\] = \[\sqrt r\] x \[\sqrt s\]

  • Rule 2:  \[\sqrt{\frac{r}{s}}\] = \[\frac{\sqrt{r}}{\sqrt{s}}\]

  • Rule 3: \[\frac{r}{\sqrt{s}}\] = \[\frac{r}{\sqrt{s}}\] X \[\frac{\sqrt{s}}{\sqrt{s}}\]

  • Rule 4: p \[\sqrt r\] ± q \[\sqrt r\] = (p ± q)\[\sqrt r\]

  • Rule 5: \[\frac{r}{p+q^{\sqrt{n}}}\]

  • Rule 6: \[\frac{r}{p-q^{\sqrt{n}}}\]


One of the methods to solve the expression is by using surds. With the rules mentioned above, you can rationalize the denominator to remove the surd. Sometimes when you’re solving the problem, it becomes necessary to solve or remove the surd. Removing the surd can be done by using one of the above-mentioned rules.  To understand it better, here’s an example - 169. First, you need to find the square root of 169. We know that 13 x 13 = 169. Here we do not have any surds. Therefore, we can say that the greatest perfect square factor of 169 is 13.


Surds Problems Examples

Question 1: What is the conjugate of a) 1+\[\sqrt 6\] + 4, b) 8\[\sqrt 3\] - 9


Solution: a) The conjugate of 1+\[\sqrt 6\] + 4 = 1-\[\sqrt 6\] + 4

      b) The conjugate of 8\[\sqrt 3\] - 9 = 8\[\sqrt 3\] + 9


Question 2: Rationalize the denominator \[\frac{3}{4+{\sqrt{2}}}\]

Solution: 

\[\frac{3}{4+{\sqrt{2}}}\] = \[\frac{3}{4+{\sqrt{2}}}\] X \[\frac{4-{\sqrt{2}}}{4-{\sqrt{2}}}\]

\[\frac{3}{4+{\sqrt{2}}}\] = \[\frac{3(4-\sqrt{2})}{(4)^{2}-(\sqrt{2})^{2}}\]

\[\frac{3}{4+{\sqrt{2}}}\] = \[\frac{12-3{\sqrt{2}}}{16-2}\]

\[\frac{3}{4+{\sqrt{2}}}\] = \[\frac{12-3{\sqrt{2}}}{16}\]


Question 3: Rationalize the denominator \[\frac{8}{1+2{\sqrt{3}}}\]

Solution: 

\[\frac{8}{1+2{\sqrt{3}}}\] = \[\frac{8}{1+2{\sqrt{3}}}\] X \[\frac{1-2{\sqrt{3}}}{1-2{\sqrt{3}}}\]

\[\frac{8}{1+2{\sqrt{3}}}\] = \[\frac{8}{1+2{\sqrt{3}}}\] = \[\frac{8(1-2\sqrt{3})}{(1)^{2}-(2\sqrt{3})^{2}}\]

\[\frac{8}{1+2{\sqrt{3}}}\] = \[\frac{8-16\sqrt{3}}{1-4(3)}\]

\[\frac{8}{1+2{\sqrt{3}}}\] = \[\frac{8-16\sqrt{3}}{1-12}\]

\[\frac{8}{1+2{\sqrt{3}}}\] = \[\frac{8-16\sqrt{3}}{-13}\]

FAQ (Frequently Asked Questions)

1. What are the Rules of Surds?

The rules of surds are: 

  • Rule 1:  = √(r*s) = √r*√s 

  • Rule 2: √(r/s) = √r/√s

  • Rule 3: r/√s = (r/√s) X (√s/√s) 

  • Rule 4: p√r ± q√r

  • Rule 5: r / (p+q√n)

  • Rule 6: r / (p-q√n)

2. What is Surds Definition?

Surds are the representation number numbers in the form of square root since these numbers cannot be a whole number or a rational number. These numbers cannot be represented as fractions too. For example, consider the value of √2. The value of  √2 is said to be recurring. The value of √2 is 1.414.