Surds and Indices-Law of Surds

In mathematics, the law of surds generally implies the square root of a given number which cannot be further simplified to a whole number or a rational number. By nature, surds cannot be perfectly represented within a fraction. Whereas indices are an order of exponents that are raised into a given number or any sort of variable. For instance take any number like 42, where 2 is said to be the index and 4 is said to be the base.

Conceptualization of Surds

Surds can be basically defined as values of roots which could not be written down as whole numbers. In mathematics; surds are generally the square root of a given number which cannot be further simplified to a whole number or a rational number. By nature, surds cannot be perfectly represented within a fraction. In order to understand the idea of surds, the focus should be done on the “law of surds”.  Let us take a numerical example for a better understanding of the “law of surds”.  Basically, irrational numbers which are not represented in the fractional form are called surds.  It can be alternatively said that “law of surds” is nothing however representations of square roots of non-rational numbers which could not be expressed in case of recurring decimals.

Different categories of Surds

In the earlier section, after having a detailed explanation about the basic concept of the nature of surds, the “law of surds” times to discuss the various types of surds that are discussed below. Each type of surds follows a different type of “Law of Surds Formula”. First 

• Pure Surds do not have any factor other than 1 such as √2, 
• Mixed surds have factors other than 1, for example 3√5, 4√6
• When two radicands of two surds are equal, then it is termed as Like suds such as 4√5 and 216√5
• When the radicands are different, the surds are known as unlike surds such as 3√5 and √3

Other surds are mainly compound and binomial surds. All these have different “laws of surds” and they tend to follow a separate “Law of surds formula” that is being applied numerically with an inclusion of some basic steps. 

Numerical application of Surds with basic fundamental steps

“Law of surds formula” applies numerically with the inclusion of some fundamental steps. These can be broadly discussed below:

Law no. 1:- 

A quadratic surd cannot be equal to sum or differences of a rational number and quadratic surd 

Such as a+√b = c+√d or a- √b = c-√d then the result will be a=c and b=d

Law no. 2:- If x and y are two different numbers, then it can be said that if root x is divided by root y the result which is being obtained can be illustrated with the help of an appropriate example. 

For instance, root 18 is divided by root 169; the result which is obtained is 3 roots 2 by 13.

Law no. 3:- By means of rationalizing the denominator by the multiplication of the numerator.

In order to understand the various “law of surds”, it is very much significant to understand all these fundamental steps of the “Law of surds formula”. If any student is found to apply the “law of surds” in a wrong way, then the whole solution turns out to be wrong. So, it is important for all the learners to keep in mind and tactfully apply the “law of surds” before proceeding into solving surds and indices.

Conclusion

It can be concluded from the above discussion that surds are basically the values of roots which could not be written down as whole numbers. In comparison, it is being said that Indices are the order of powers or exponents of a given specific value. In mathematics, surds are generally the square root of a given number which cannot be further simplified to a whole number or a rational number. It can be stated that surds are being used in practical and real life in making sure that essential numerical calculations in general are simple and precise. Generally, surd is an overall expression which encompasses square, cubical root or any other particular symbol. These are in general used in writing non rational numbers on simple grounds since “law of surds” is not written appropriately within decimal form.