What does the power rule of exponents state when multiplying exponents?

The power rule of exponents states that when you multiply two expressions with the same base, you add the exponents. This means that if you have a base ( a ) raised to an exponent ( m ) and another expression with the same base ( a ) raised to an exponent ( n ), the operation can be expressed as:

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

This fundamental property simplifies calculations involving exponents significantly and is crucial for algebraic operations involving polynomial expressions. For example, if you multiply ( x^3 ) and ( x^2 ), applying the power rule allows you to combine the expressions directly to get ( x^{3+2} = x^5 ). Understanding this rule is essential for successfully manipulating and simplifying equations in a variety of mathematical contexts.