In an, a is the base and n the index. For positive integer n, it represents n equal factors. Index laws follow by counting, grouping or cancelling these factors. **Operation** | **Law** Multiply powers with the same base | aman=am+n Divide powers with the same nonzero base | aman=am-n Raise a power to a power | (am)n=amn Raise a product to a power | (ab)n=anbn Raise a quotient to a power, b 0 | (ab)n= anbn Example (2x3)2x4x2= 4x74x2=x5, for x 0. These are multiplication and division rules. x3+x4 cannot be simplified to x7; adding terms does not concatenate their factors.

In \(a^n\), \(a\) is the base and \(n\) the index. For positive integer \(n\), it represents \(n\) equal factors. Index laws follow by counting, grouping or cancelling these factors.

\(\dfrac{(2x^3)^2x}{4x^2}=\dfrac{4x^7}{4x^2}=x^5\), for \(x\ne0\).

These are multiplication and division rules. \(x^3+x^4\) cannot be simplified to \(x^7\); adding terms does not concatenate their factors.

Operation	Law
Multiply powers with the same base	\(a^ma^n=a^{m+n}\)
Divide powers with the same nonzero base	\(\dfrac{a^m}{a^n}=a^{m-n}\)
Raise a power to a power	\((a^m)^n=a^{mn}\)
Raise a product to a power	\((ab)^n=a^nb^n\)
Raise a quotient to a power, \(b\ne0\)	\(\left(\dfrac ab\right)^n=\dfrac{a^n}{b^n}\)

Positive integer powers and index laws

Positive integer powers and index laws