Exponents, including the strange ones

Walk the powers of 2 downwards past zero and the meaning of $2^{0}$ and $2^{-3}$ is forced on you.

Writing $2 \times 2 \times 2 \times 2$ as $2^{4}$ is only shorthand, and if that were all exponents were, they would need no lesson. The interesting question is what the shorthand ought to mean when the little number is zero, or negative, and there is nothing left to multiply.

Rather than being told, watch the pattern walk down past the point where the shorthand runs out.

Reading upwards, every step multiplies by 2: 8, 16, 32, 64. So reading downwards, every step must divide by 2: 64, 32, 16, 8, 4, 2, and then 1. That 1 is $2^{0}$, and it did not come from a rule somebody invented. It is where the halving lands.

Keep halving. After 1 comes $\frac{1}{2}$, which the output calls 0.5, and that is $2^{-1}$. Then $\frac{1}{4}$, which is $2^{-2}$. A negative exponent is not a negative number and it never produces one; it is a division.

The definitions the pattern forces

For any real number $a$ that is not zero, and any positive whole number $n$:

Zero is excluded because $0^{0}$ has no single sensible value and $0^{-1}$ would be a division by zero. Every other real base is fine.

The five laws

All five come from the same observation: an exponent counts how many copies of the base are being multiplied, and multiplying two piles of copies together makes one bigger pile.

The laws hold for negative exponents too, which is the real reason the definitions above were chosen. Test a few at once and watch the two columns agree.

Two traps worth meeting on purpose

The first is about what a minus sign in front is attached to. An exponent binds more tightly than a minus sign, so $-3^{2}$ means "take $3^{2}$, then negate it", which is $-9$. Only $(-3)^{2}$ squares the negative number, giving $9$.

The second trap is the belief that an exponent spreads across a sum the way it spreads across a product. It does not, and one pair of numbers is enough to prove it.

A worked example, every step

Simplify $\dfrac{\left(2x^{3}y^{-2}\right)^{-2}}{4x^{-1}y}$ and leave no negative exponents in the answer.

1. Deal with the bracket first. The outer $-2$ hits every factor inside: $2^{-2}$, then $\left(x^{3}\right)^{-2}$, then $\left(y^{-2}\right)^{-2}$.
2. $2^{-2} = \frac{1}{4}$. For the letters, multiply the exponents: $x^{3 \cdot (-2)} = x^{-6}$ and $y^{(-2)\cdot(-2)} = y^{4}$.
3. So the numerator is $\frac{1}{4}x^{-6}y^{4}$.
4. Now divide by $4x^{-1}y$. The plain numbers give $\frac{1}{4} \div 4 = \frac{1}{16}$.
5. Dividing subtracts exponents: $x^{-6-(-1)} = x^{-6+1} = x^{-5}$, and $y^{4-1} = y^{3}$.
6. That leaves $\frac{1}{16}x^{-5}y^{3}$.
7. Finally clear the negative exponent by moving $x^{-5}$ to the bottom as $x^{5}$: the answer is $\dfrac{y^{3}}{16x^{5}}$.

Scientific notation

A number is in scientific notation when it is written as one digit, then a decimal point, then the rest, all multiplied by a power of ten. Formally it is $a \times 10^{n}$ where $n$ is an integer and $1 \le |a| < 10$. The mass of the Earth is $5.97 \times 10^{24}$ kilograms and the mass of an electron is $9.11 \times 10^{-31}$ kilograms.

The exponent says how far to move the decimal point: positive moves right and makes the number big, negative moves left and makes it small. Most languages write it with an e, so 5.97e24 is the same thing typed on a keyboard.

Multiplying two such numbers is one law of exponents and one ordinary multiplication. Take $(3.2 \times 10^{5})(4 \times 10^{-9})$. Multiply the fronts: $3.2 \times 4 = 12.8$. Add the exponents: $5 + (-9) = -4$. That gives $12.8 \times 10^{-4}$, which is not yet in scientific notation because 12.8 is bigger than 10. Move the point one place left and add one to the exponent: $1.28 \times 10^{-3}$.

Dividing works the same way with the other two laws: divide the fronts, and subtract the exponents. Here is one all the way through.

1. The problem is $\dfrac{6 \times 10^{8}}{1.5 \times 10^{-3}}$. Split it into the fronts and the powers: $\dfrac{6}{1.5} \times \dfrac{10^{8}}{10^{-3}}$.
2. Divide the fronts: $6 \div 1.5 = 4$.
3. Divide the powers by subtracting exponents: $8 - (-3) = 8 + 3 = 11$, so that part is $10^{11}$. Subtracting a negative adds, which is where this step goes wrong most often.
4. Put them back together: $4 \times 10^{11}$.
5. Check that it is in scientific notation: the front is 4, and $1 \le 4 < 10$, so nothing needs moving.
6. Sanity check the size. The top is a few hundred million, the bottom is about a thousandth, and dividing by a thousandth multiplies by a thousand, so an answer in the hundreds of billions is the right order of magnitude.