The 8 Laws Of Indices In Maths Explained

Index (indices) in Maths is the exponent which is raised to a number. For example, in number 4², 2 is the index or power of 4. The plural form of index is indices. Also, a number of the form xⁿ where x is a real number, x is multiplied by itself n times i.e. xⁿ = x*x*x*x*x*——(n times). The number x is called the base and the super script n is called the index or power or exponent. In this article, you will learn the laws of the indices along with formulas and solved examples.

Index laws are what you have to understand if you want to gain mastery in solving any problem in mathematics.

Read: Steps to get good grade in math

8 Laws of Indices

1^st law

Any base variable raise to zero (0) is one (1) i.e. A⁰ = 1. For example, 2⁰ = 1

2^nd Law

If a base variable is raised to a negative number, then it will be equal to inverse of the base variable raised to a positive number i.e.

A^-m = 1/A^m

3^rd Law

If a base number (which is a fraction) is raised to a number (m), then it will be equal to the numerator raised to the number (m) and the denominator raised to the same number (m) i.e.

(A/B)^m = A^m/B^m

Example, (2/3)² = 2²/3² = 4/9

4^th Law

If a base number is raised to a number (m) and multiply a base number of the same value raised to a number (n), then it will be equal to the base number raised to the sum of the exponents (m + n) i.e.

A^m X Aⁿ = A^m+n

Example, y⁵ x y⁴ = y⁵⁺⁴ = y⁹

5^th Law

If a base number is raised to a number (m) and is divided by a base number of the same value raised to a number (n), then it will be equal to the base number raised to the difference between the exponents (m – n) i.e.

A^m -:- Aⁿ = A^m-n

Example, 5⁴ -:- 5² = 5^4-2 = 5² = 5 x 5 = 25

6^th Law

If a base variable is raised to a number (which is a fraction (x/y)), then it will be equal to the yth root of the base number raised to x i.e.

A^x/y = ^y√A^x

Example, 27^2/3 = ³√27² = 3² = 9

7^th Law

If a base varaible is raised to a number (n) and the entire number raised to power (m), then it is equal to the base number raised to the multiplication of the two exponents (m*n) i.e.

(A^m)ⁿ = A^mn

Example, (3³)² = 3^3*2 = 3⁶ = 3*3*3*3*3*3 = 729

8^th Law

When two base variables with different bases, but same indices are multiplied together, we have to multiply the two bases and raise the same index to multiplied variables i.e.

Aⁿ x Bⁿ = (A.B)ⁿ

Example, 3² x 2² = (3*2)² = 6² = 36

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