Surds

For understanding surds, we should first have basic knowledge about rules of exponents. A surd is a bit more advanced form of an indice. So, learning about surds is what this wiki is about.

First a basic question. What is $\sqrt{4}$? Obviously the answer is $2$ as its perfect value can be found easily. Now, what is $\sqrt{5}$? After long division we get is as $2.236\cdots$. But that is not the exact value. Such numbers are called surds.

So, we can define surds as any root of such a number whose exact value can't be found.

If $x$ is a rational number and its $y^{th}$ root, that is $x^{\tfrac{1}{y}}$ is irrational, then $\sqrt[y]{x}$ is a surd which has order $y$.

Identify the order of the surd $\sqrt[10]{1001}$.

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$$\begin{aligned} \sqrt[10]{1001} & = {1001}^{\tfrac{1}{10}} \end{aligned}$$
$\text{So, we see that the given number is the 10th root of 1001}$
$\therefore \text{The order of the surd is 10}.$

The following are the various types of surds:-
1) $\text{Simple Surds}$: Surds which consists of only one term is called a surd. For example, $5\sqrt{5}$, $\sqrt{7}$, etc.
2) $\text{Similar Surds}$: If the surds having the same common surds factor, or in other words, they are different multiples of the same surd, then they are called similar surds.
3) $\text{Pure Surds}$: Surds which are wholly irrational are called pure surds. Eg:- $\sqrt{3}$.
4) $\text{Mixed Surds}$: Surds which are not wholly irrational and can be expressed as a product of a rational number and an irrational number is called mixed surd.
5) $\text{Binomial Surds}$: A surd which is made of two other surds is called binomial surd.
6) $\text{Compound Surds}$: An expression which is the addition or subtraction of two or more surds is called compound surd.

If two binomial surds are such that only the sign connecting the individual terms are different, then they are said to be conjugate of each other. If theae surds are quadratic, then their product would always be rational. So, in case of a binomial quadratic surd, we use its conjugate as its rationalizing factor.

What is the conjugate of $6\sqrt{7} + 3\sqrt{5}$?

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$6\sqrt{7} - 3\sqrt{5}$

Rationalize the denominator of $\dfrac{1}{8\sqrt{11} - 7\sqrt{5}}$?

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Conjugate of $8\sqrt{11} - 7\sqrt{5} = 8\sqrt{11} + 7\sqrt{5}$
Therefore multiplying the conjugate in the numerator and denominator of the given fraction.
$$\begin{aligned} \dfrac{1 × (8\sqrt{11} + 7\sqrt{5})}{(8\sqrt{11} - 7\sqrt{5})(8\sqrt{11} + 7\sqrt{5})} & = \dfrac{(8\sqrt{11} + 7\sqrt{5})}{{(8\sqrt{11})}^2 - {(7\sqrt{5})}^2}\\ & = \dfrac{(8\sqrt{11} + 7\sqrt{5})}{704 - 245}\\ & = \boxed{\dfrac{(8\sqrt{11} + 7\sqrt{5})}{459}} \end{aligned}$$

Find the square root of $5$ + $2\sqrt{6}$.

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Solution to be added.