### Exponents & More

Topics within Exponents & More

## Properties

Hopefully, those practice problems gave you some intuition for how exponents work. Exponents have several laws, or properties. I've listed them below. If you've done the practice problems, you are already familiar with most of these laws because many of the practice problems are just examples of these laws in disguise.

Product of Powers: $$x^a\cdot x^b=x^{a+b}$$

Quotient of Powers: $$\frac{x^a}{x^b}=x^{a-b}$$

Power of a Product: $$x^a+y^a=\left(xy\right)^a$$

Power of a Quotient: $$\frac{x^a}{x^b}=x^{a-b}$$

Power of a Power: $$\left(x^a\right)^b=x^{ab}$$

Identity Exponent: $$x^1=x$$

Zero Exponent: $$x^0=1$$

Negative Exponent: $$x^{-a}=\frac{1}{x^a}$$

Fractional Exponent: $$x^{\frac{a}{b}}=\sqrt[b]{x^a}=\left(\sqrt[b]{x}\right)^a$$

Your homework is to prove these laws to yourself. In other words, I want you to understand why these laws are true. I encourage you to try out examples and to use the first few laws to prove the other laws. If you've done the practice problems, you only have the last two laws left. Good luck!