Exponentiation is repeated multiplication: xn means multiply x by itself n times.
For example: 23 = 2 x 2 x 2 = 8.
Base (x): The number being multiplied.
Exponent (n): How many times the base is used as a factor.
Zero exponent: x0 = 1 (for x ≠ 0).
Negative exponent: x-n = 1/xn.
Fractional exponent: x1/n = √n(x). The denominator is the root index.
Product rule: xa x xb = xa+b.
Quotient rule: xa / xb = xa-b.
Power of a power: (xa)b = xab.
Power of a product: (xy)n = xn x yn.
Square (n=2): x² — the area of a square with side x.
Cube (n=3): x³ — the volume of a cube with side x.
Square root (n=1/2): x1/2 = √x.
Cube root (n=1/3): x1/3 = ³√x.
Science and engineering: Scientific notation uses powers of 10. Compound interest uses exponential growth.
Computer science: Binary numbers use powers of 2. Algorithm complexity is expressed as powers of n.
Finance: Compound interest: A = P(1 + r/n)nt.
Geometry: Area scales with the square of linear dimensions; volume scales with the cube.