## Exponents in Algrebra

An index, power or exponent is a way of shortening expressions such as 3 × 3 × 3 × 3 × 3 × 3 × 3 to 37

4p5 is a short way of writing 4 × p × p × p × p × p.

The 4 is called the coefficient , the p is called the base or variable and the 5 is called theindex, power or exponent.

### Index Facts

The value of a variable to the power of 0 is always 1.

e.g. x0 = 1

A variable to the power of 1 is the variable itself.

e.g. x1 = x

Indices (plural of index) obey certain rules when they are being multiplied and divided.

### Multiplication

When multiplying numbers or variables with indices, add the indices together.

The bases must be the same number or variable.

 Method Examples When multiplying add the indices. a3 × a4 = a 3 + 4 = a7 Coefficients are multiplied as normal. 3p3 × 4p2 = 12p5

### Division

When dividing numbers or variables with indices, subtract the indices.

The bases must be the same number or variable.

 Method Examples When dividing subtract the indices. p5⁄p3 = p5−3 = p2 Coefficients are divided as normal. 12x6⁄4x2 = 3x6−2 = 3x4

### Powers in Brackets

When a term inside a bracket is raised to a power, for example (x2)3 then the two powers aremultiplied .

 Method Examples For a power to another power multiply the indices. (p2)3 = p2×3 = p6 Coefficients are calculated as normal exponents. (2q3)4 = 16q12

### Roots

When the root of a variable to an exponent is taken, the exponent is dividedi.e. For the squareroot, divide the exponent by 2.

 Method Examples For the root of an exponent divide the index. √p6 = p6÷2 = p3 Coefficients are calculated as normal roots. √(16p8) = 4p8÷2 = 4p4