## Indices

4p5 is a short way of writing 4 × p × p × p × p × p. The 4 is called the coefficient , the p is called thebase or variable and the 5 is called the index, power or exponent.

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

### Rules of Indices

Index of one

A number or variable to an index of 1, is equal to itself.

e.g. 41 = 4

y1 = y

Multiplication

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

The bases must be the same number or variable.

e.g        a3 × a4 = a 3 + 4 = a7

because a3 × a4 = (a × a × a) × (a × a × a × a) = a7

Division

When dividing numbers or variables with indices, subtract the indices. The bases must be the same number or variable. Two indices

When raising a number or variable with an index to another index, multiply the two indices together.

e.g. (a3)= a3x2 = a6

because (a)2 = (a × a × a) × (a × a × a) = a6

Index of zero

Any number or variable to the index of 0 is equal to 1. Note: 00 is undefined. i.e. it does not exist. Negative index

A number or variable raised to a negative index is equal to the reciprocal of the number or variable to the same positive index. ### Summary

 Rules of indices ax x ay = ax + y Multiplying (add the indices) Dividing (subtract the indices) (ax ) y = axy Powers (multiply the indices) a1 = a Power of one (stays same) a0 = 1, a ≠ 0 Power of zero (always equals one) Negative index (take the reciprocal)

### Powers of Ten

With a number multiplied by a power of 10, the index indicates the direction and the number of places the point must be moved.

e.g.    1.30 × 102 = 1.30 × 100 = 130 (point moves 2 places right)

0.0562 × 103 = 0.0562 × 1000 = 56.2 (point moves 3 places right)

7.64 × 10 -2 = 7.64 ÷ 100 = 0.0764 (point moves 2 places left)

### Expressions involving Indices

To simplify algebraic expressions involving indices, use the rules of indices.

Simplify the signs first, numbers next and then the similar variables. 