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
Summary of 
Rules of Indices
Powers of Ten
Expressions involving indices


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


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


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.





Rules of indices
ax x ay = ax + y

Multiplying (add the indices)


Dividing (subtract the indices)

(ay = 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.