INDICES

FOR BETTER UNDERSTANDING
THE CARET SIGN (^) MEANS RAISE TO POWER

Index (indices) in Maths is the power or exponent which is raised to a number or a variable. The plural form of index is indices. In algebra, we come across constants and variables. The constant is a value which cannot be changed. Whereas a variable quantity can be assigned by any number or we can say its value can be changed. In algebra, we deal with indices in terms of numbers. Let us learn the laws/rules of the indices along with formulas and solved examples.

Index (raise to power) Definition

A number or a variable may have an index. Index of a variable (or a constant) is a value that is raised to the power of the variable. The indices are also known as powers or exponents. It shows the number of times a given number has to be multiplied. It is represented in the form:

a^m= a × a × a ×……× a (m times)

Here, a is the base and m is the index.

The index says that a particular number (or base) is to be multiplied by itself, the number of times equal to the index raised to it. It is a compressed method of writing big numbers and calculations.

Example:  2³ = 2 × 2 × 2 = 8

In the example, 2 is the base and 3 is the index.

Laws of Indices

There are some fundamental rules or laws of indices which are necessary to understand before we start dealing with indices. These laws are used while performing algebraic operations on indices and while solving the algebraic expressions, including it.

Rule 1: If a constant or variable has index as ‘0’, then the result will be equal to one, regardless of any base value.

a⁰ = 1

Example: 5⁰ = 1, 12⁰ = 1, y⁰= 1

Rule 2: If the index is a negative value, then it can be shown as the reciprocal of the positive index raised to the same variable.

a-² = 1/a²

Example: 5-¹ = ⅕, 8-³=1/8³

Rule 3: To multiply two variables with the same base, we need to add its powers and raise them to that base.

a² × a³ = a²+³

Example: 5²×5³ = 5²+³ = 5⁵

Rule 4: To divide two variables with the same base, we need to subtract the power of denominator from the power of numerator and raise it to that base.

A³ ÷A² = A³-²  =A¹

Example: 10⁴/10² = 10⁴-² = 10²

Rule 5: When a variable with some index is again raised with different index, then both the indices are multiplied together raised to the power of the same base.

(A²)³ = A²×³

Example: (8²)³= 8²×³= 8⁶

Rule 6: When two variables with different bases, but same indices are multiplied together, we have to multiply its base and raise the same index to multiplied variables.

A^p×B^p= (AB)^p

Example: 3²×5² = (3 x 5)² = 15²

Rule 7: When two variables with different bases, but same indices are divided, we are required to divide the bases and raise the same index to it.

a^p/b^p = (a/b)p

Example: 3²/5² = (⅗)2

Rule 8: An index in the form of a fraction can be represented as the radical form.

A^p/q = q√A^p

Example: 6^½ = √6

If you can solve this, then you have no problem
Just an example
Find the value of 272/3.

Solution: 27^2/3

= 3√27²

= 3√3^³(²)
Here The index power of 3 will break the cube root, therefore
You have
3²which will equal
= 9