# 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 4^{2}, 2 is the index or power of 4. The plural form of index is indices. Also, a number of the form x^{n} where x is a real number, x is multiplied by itself n times i.e. x^{n} = 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.

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**8 Laws of Indices**

**1 ^{st} law**

Any base variable raise to zero (0) is one (1) i.e. A^{0} = 1. For example, 2^{0} = 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} = 2^{2}/3^{2} = 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 A^{n} = A^{m+n}

Example, y^{5} x y^{4} = y^{5+4} = y^{9}

**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} -:- A^{n} = A^{m-n}

Example, 5^{4} -:- 5^{2} = 5^{4-2} = 5^{2} = 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}√27^{2} = 3^{2} = 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, (3^{3})^{2} = 3^{3*2} = 3^{6} = 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.

A^{n} x B^{n} = (A.B)^{n}

Example, 3^{2} x 2^{2} = (3*2)^{2} = 6^{2} = 36

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