Simplifying powers and matching bases

Choose the index law that matches the operation, then leave a positive-index form when requested. Simple equations with the unknown in an index can be solved when both sides can be expressed in one positive base other than \(1\). Equal powers then have equal indices.

Example

\(\begin{aligned}2^{x+1}&=32\\2^{x+1}&=2^5\\x+1&=5\\x&=4\end{aligned}\)

Check in the original equation: \(2^{4+1}=32\).

With positive \(a\), \(a^{\frac12}a^{\frac32}=a^2\). By contrast \((a^{\frac12})^3=a^{\frac32}\): multiplying powers adds indices, while raising a power multiplies them.