Laws of indices
In a nutshell
Indices, or an index, refers to the power on a number or variable. There are different rules which show how to manipulate expressions with indices, including how to multiply or divide indices or how to calculate bracketed indices. This set of rules is known as the index laws.
Multiplying with indices
If two numbers with powers are multiplied together, it is possible to simplify the power, as long as the bases of the two numbers are the same. When two numbers with the same base, but different powers are multiplied, add the powers. Use the rule:
am×an=am+n
Example 1
Simplify 2x5×3x3.
The terms are multiplied, so add the powers.
2x5×3x3=6x5+3=6x8
Dividing with indices
If two numbers with powers are divided, it is possible to simplify the power, as long as the bases of the two numbers are the same. When two numbers with the same base, but different powers are divided, subtract the powers. Use the rule:
am÷an=am−n
Example 2
Simplify 15x5÷3x2.
The terms are divided, so subtract the powers.
15x5÷3x2=5x5−2=5x3
Bracketed indices
Bracketed indices are expressions where a term with a power is raised to another power, for example (x3)4. Here, x3 is multiplied by itself four times. Bracketed indices can be simplified by multiplying the powers. Use the rule:
(am)n=amn
Example 3
Simplify (2x3)2.
Multiply the indices.
(2x3)2=22×x3×2=4x6
Tip: Remember that the coefficient is also raised to the power!