In this video, we present and prove five properties of exponents via a simple definition and our prior properties of real numbers. The proofs presented in the video do not represent rigorous mathematical arguments. Instead they area means to mathematical reasoning and are given primarily to aid remembering the rules for manipulating exponents.

In particular, we use a definition of exponent that can only apply to exponents, which are natural numbers. However, the properties given here actually apply to any real number exponent. Our extension to all integer exponents comes via the assumption that these rules hold, and without that assumption we would not be able to proceed.

So, just as we view multiplication as repeated addition, we view exponents as repeated multiplication. These definitions are historical, but do not properly define the operations fully, as multiplication and exponents apply to all real numbers, despite our simplified definitions.

When we speak of an exponent, we say that the expression above is 3 to the 5th power or 3 to the 5. The number 3 is called the base and 5 is called the exponent.