When multiplying powers of the same base, what do you do with the exponents?

When multiplying powers that share the same base, the correct action is to add the exponents. This rule stems from the definition of exponents themselves. For example, if you have a base ( a ) and two exponents ( m ) and ( n ), the expression ( a^m \times a^n ) can be simplified to ( a^{m+n} ).

This addition of exponents can be understood through the concept of repeated multiplication. If you consider ( a^m ) as ( a ) multiplied by itself ( m ) times and ( a^n ) as ( a ) multiplied by itself ( n ) times, when you multiply these two expressions together, you're essentially combining all instances of ( a ) being multiplied. Thus, you have ( a ) multiplied by itself ( m+n ) times, which leads to the conclusion that ( a^m \times a^n = a^{m+n} ).

Understanding this fundamental rule is essential for simplifying expressions and solving problems that involve exponents, and it serves as a building block for more advanced algebraic concepts.