2.8 Summary
Key Points
- n^a means n multiplied by itself a times; n is the base and a the exponent. Reading example: 5^4 = 625, “five to the power four.”
- Exponential growth is multiplicative (each step multiplies); linear growth is additive (each step adds). Folding foil 46 times overshoots the Moon, that is exponential growth at full throttle.
- The laws of exponents (valid for any integers, with n\ne 0): n^a\times n^b = n^{a+b}, \quad n^a\div n^b = n^{a-b}, \quad (n^a)^b = (n^b)^a = n^{ab}, m^a\times n^a = (mn)^a, \qquad \frac{n^a}{m^a} = \left(\frac{n}{m}\right)^a.
- The zero exponent: n^0 = 1 for n\ne 0. Negative exponents: n^{-a} = \dfrac{1}{n^a}.
- Scientific notation records any number as x\times 10^{y} with 1\le x < 10 and integer y; the exponent y carries the size and matters most.
- Powers of ten give us a grip on huge quantities, from a lakh (10^5) and arab (10^9) to a googol (10^{100}).