Surds and Indices – Law of Indices

Indices are the plural form of index. Indices can be stated as the numerical power or exponent of a number that is raised to some number or variable. In algebra, indices are widely used to solve algebraic sums. This article revolves around laws of indices and laws of indices formula. Laws of indices help to solve algebraic problems based on numeric constants and variables. Laws of indices formula state a particular way of solving algebraic problems in mathematics. 

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Mathematical concept of indices

In algebra, a numeric constant or variable is raised to some exponential power. Laws of indices help students to solve particular problems through specific laws of indices. Indices of a numeric constant or a variable are considered as a value that is raised to some exponential power of variables. In this regard, indices are also considered as power of exponents. Indices represent the number of times that a given number has been multiplied by itself or raised to some power. Indices can be represented by mathematical equations.

am = a x a x a x a x a ………… x a ( up to m times)

Where “a” is the base number and m is considered as indices. In this way, indices “m” represent that the base “a” is multiplied by itself to m times. It is justified by a numerical example 32

25 = 2 x 2 x 2 x 2 x 2 = 32

Here, 2 is base and 5 are indices.

Laws of indices

It requires some basic fundamental rules or laws of indices while dealing with numerical or algebraic problems related to indices. To get a better understanding of indices requires conceptualising laws of indices and formulas related to laws of indices. Laws of indices are the rules that are used for simplifying expressions involving power of a particular base number. Laws of indices also provide rules for solving and simplifying expressions and equations involving power of the same or different base. 

Kinds of indices and their application

It is found that indices are of several kinds that follow certain laws of indices. Kinds of indices include multiplying, dividing, negative and fractional power, power of 0, brackets. This article reveals each kind of indices and justifies it with suitable examples. Indices are applied through several rules based on laws of indices. Application of indices is discussed with the help of some rules and justified with some examples.

• Multiplying indices

Here the same base will lead to addition of power.

am x an = am+n

Example: 23 x 24 = 23+4 = 27

• Dividing indices

In dividing indices, power will be subtracted

am ÷ an = am-n

Example: 56 ÷ 52 = 56-2 = 54

• Bracket indices

The power will be multiplied if brackets are used in indices.

(am)n = amxn

Example: (82)5 = 82×5 = 810

• Negative indices

Negative indices will be flip or written in reciprocal form to make it positive

a-m = 1/ am

Example: 9-1 = 1/9; 6-3 = 1/63

• Fractional indices

If the power of indices is given in fractional form the denominator is considered as the root of the number and it is raised to the power of the numerator. 

xa/b=(b√x)a

64/3=(∛6)4

• Power of 0

Any indices that are raised to the power of 0 then the value will be equal to 1

a0=1

12530=1, 70=1

• Rule 1: When two variables have different bases but indices are multiplied together. In this regard, bases are multiplied together and they will be raised to given power. 

ap.bp = (ab)p

Example: 45 x 65 = (4×6)5 = 245

• Rule 2: If indices are in fraction, then it will be represented in the radical forms. 

ap/q = q√ ap

Example: 61/2 = √6

Conclusion

It is to be concluded that indices are the number or variable that are raised to another number of variables exponentially. It is observed that indices are the plural form of index. Laws of indices are also known as exponents or power. It is observed during this study that indices not only help in solving mathematical problems related to algebra whereas, it also helps in calculating several measurements in real life. In comparison with surds, indices are the power of exponent while surd is the root form of any number or variable. Various laws of indices formula are used to solve problems related to indices.