Laws of exponents
Powers are shorthand for long products. How do you multiply, divide and raise them without writing the products out, and why do the rules work?
Where you are 3 inside
What this area is about
A power such as 2^{10} counts factors. The laws of exponents are what that counting gives when powers are combined: multiplying lines up two rows of factors, so exponents add; dividing cancels factors, so exponents subtract; a power of a power repeats a row, so exponents multiply. None of them needs memorizing apart from that reason, and the same reason decides what a zero or negative exponent has to mean.
How these fit together
The first rule. Multiplying and dividing powers with the same base gives a^m \cdot a^n = a^{m+n} and, for a \ne 0, a^m \div a^n = a^{m-n}.
Extending it. Powers of powers, products and quotients gives (a^m)^n = a^{mn}, and shows how a power spreads over a product or a fraction, as in (5x)^3.
New exponents. Zero and negative exponents defines a^0 = 1 and \displaystyle a^{-n} = \frac{1}{a^n}, for a \ne 0, as the only meanings that keep the dividing rule true.
A path through it
Start from exponent notation. Learn the product and quotient rule first; the power-of-a-power rule and zero and negative exponents both follow from it, in either order.
These laws make scientific notation, multiplying monomials and rational exponents possible.