Online Mathematics Book

Laws of Exponents & Indices & Powers

The laws of exponents, or laws of indices, or even power laws are all the rules we have to know on order to do operations with numbers raised to a power.

In this section we learn the rules for all the basic operations we need to know how to do with powers of numbers:

  • addition & subtraction
  • multiplication
  • division
  • raising to a power

Addition & Subtraction Laws

No simplication can be made when adding, or subtracting, two exponentials.
In mathematical terms that's \[a^m + a^n = a^m + a^n\]

and
\[a^m - a^n = a^m - a^n\]

Note: as such this is not much of a formula, it is nevertheless important as it helps us avoid making

Example

Multiplication Laws

Same Base

\[ a^m \times a^n = a^{m+n}\]

Different Base

\[a^m\times b^n = a^m \times b^n\] No simplication can be made.
exception
If exponents are equal \(m=n\) \[a^m\times b^m = \begin{pmatrix}a\times b \end{pmatrix}^m\]

Example

\(3^2\times 3^5 = 3^{2+5} = 3^7\)

Example

Exercise

Answers Without Working