Scientific notation

The form is $a \times 10^n$ with $1 \le |a| < 10$. The exponent $n$ is the order of magnitude, so $6.67 \times 10^{-11}$ and $9.0 \times 10^9$ are twenty orders of magnitude apart without you counting a single zero.

The exponent arithmetic is where marks go:

• Multiplying, multiply the front numbers and add the exponents: $(3 \times 10^8)(2 \times 10^{-5}) = 6 \times 10^3$.
• Dividing, divide the fronts and subtract the exponents.
• Raising to a power, raise the front and multiply the exponent: $(3 \times 10^8)^2 = 9 \times 10^{16}$, not $9 \times 10^{10}$.
• Adding or subtracting, rewrite both terms with the same exponent first. $3 \times 10^{-3} + 4 \times 10^{-4}$ is $3.4 \times 10^{-3}$, not $7 \times 10^{-7}$.

The AP constants are all written this way: $G = 6.67 \times 10^{-11}$, the Coulomb constant $k = 9.0 \times 10^9$, the elementary charge $e = 1.60 \times 10^{-19}\ \text{C}$ and $c = 3.00 \times 10^8\ \text{m/s}$.

One calculator habit is worth building before exam day. Enter the exponent with the EE or EXP key rather than keying the multiplication and the power separately, and bracket any negative exponent. Dividing by $2 \times 10^{-3}$ without brackets is where a factor of a thousand quietly disappears.