Exponents and roots
How do you write a number multiplied by itself many times, and how do you get back to the number you started from?
Where you are 5 inside
What this area is about
An exponent counts repeated factors: 3^4 = 3 \times 3 \times 3 \times 3. Combining powers by counting factors gives the laws of exponents, and running a power backward gives a root. A fractional exponent then joins the two, so that, for a \ge 0, \displaystyle a^{1/2} = \sqrt{a}.
How these fit together
Notation. Exponent notation names the base and the exponent and the powers worth knowing on sight.
Results. The laws of exponents add exponents when powers with the same base multiply, subtract them when such powers divide (for a nonzero base), multiply them for a power of a power, and decide what zero and negative exponents must mean.
Undoing. Square and cube roots undo squaring and cubing, and settle how many roots a number has, if any.
Applications. Scientific notation writes a very large or very small number as a \times 10^{n} with 1 \le a < 10 and n an integer. Rational exponents define a^{m/n} through roots.
A path through it
Exponent notation comes first. The laws come before scientific notation; roots can come before or after the laws, but both come before rational exponents.
Next come the Pythagorean theorem, exponential functions, multiplying monomials and simplifying square roots.