Unit 10 · Lesson 10.1

Exponent rules (including zero & negative exponents)

Exponents are just a shorthand for "multiply this by itself a few times." That's the whole idea, and every rule in this lesson comes straight out of it. The trick is to never let the shorthand pull you away from what it's counting.

When you're unsure what a rule should be, you can always write the factors out and count them. That habit, more than any memorized rule, is what keeps you out of trouble here.

These rules are the grammar of nearly every expression ahead: scientific notation in the next lesson, multiplying polynomials later in this one, and the two units after this. The way to make them stick is to see why each one is what it is, so that's how we'll build them.

Here's the first one, built from the ground up. Take x² · x³. Don't reach for a rule yet. Just write out what each piece means and count: $$x^2\cdot x^3 = (x\cdot x)(x\cdot x\cdot x) = x^5$$ Line up the x's: two of them, then three more, five in all. When you multiply two powers of the same base, you're pooling all their factors into one pile, so the exponents add. Hold onto that single picture: pool the factors, count them. Once that habit is in place, the add-versus-multiply mix-up later in this lesson has much less to grab onto.

Division works the same way, read in reverse. Take x⁵ over x²: $$\frac{x^5}{x^2} = \frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x} = x^3$$ Two factors on the bottom pair off with two on the top, and a matched pair goes to one. A factor over itself is just 1. Three factors are left over on top, so x³. Dividing like bases is "how many factors are left after the pairs go to one," which is the same as subtracting the exponents.

Now a different move that looks similar but isn't. What about (x²)³, a power raised to a power? Write that out too: $$(x^2)^3 = x^2\cdot x^2\cdot x^2 = x^6$$ Three copies of x², so 2 + 2 + 2 = 6. Here the exponents multiply.

Sit with the contrast for a second, because it's the heart of the lesson. x² · x³ is two separate powers multiplied, so you add (2 + 3 = 5). (x²)³ is one power taken three times, so you multiply (2 · 3 = 6). Same-looking, opposite arithmetic. When you can feel which situation you're in, you've got the lesson.

Gathered up, here are the rules these pictures give you: $$x^a\cdot x^b=x^{a+b}\qquad\frac{x^a}{x^b}=x^{a-b}\qquad (x^a)^b=x^{ab}\qquad (xy)^a=x^a y^a\qquad x^0=1$$

That fourth one says a power on a product reaches every factor inside: (xy)ᵃ hands the exponent to the x and to the y alike. There's a matching version for a quotient, (x/y)ᵃ = xᵃ/yᵃ, with the exponent reaching the top and bottom both. You won't need it for this unit's problems, but it's the natural companion if you ever wonder.

Where x⁰ = 1 and negative exponents come from

The last two rules look like things you'd just have to memorize. You don't, and seeing why frees you from getting them backward.

Start with x⁰. Take any power over itself, say x³/x³, and read it two ways. By the quotient rule, you subtract the exponents: x³⁻³ = x⁰. But anything divided by itself is plainly 1. Both readings describe the same quantity, so: $$\begin{aligned} &\frac{x^3}{x^3}=x^{3-3}=x^0 \quad\text{and}\quad \frac{x^3}{x^3}=1 \\ &\Rightarrow\; x^0=1 \end{aligned}$$ So x⁰ = 1 isn't a decree someone made up; it's forced. The rule you already trust leaves no other option.

Negative exponents come from the same move, just with more on the bottom than the top. Take x² over x⁵, and again read it two ways. The quotient rule gives x²⁻⁵ = x⁻³. Counting factors gives something you can see: two factors on top pair off with two on the bottom and go to one, leaving three factors stranded on the bottom, which is 1/(x³). Same quantity, two descriptions: $$\begin{aligned} &\frac{x^2}{x^5}=x^{2-5}=x^{-3}\quad\text{and}\quad\frac{x^2}{x^5}=\frac{1}{x^3} \\ &\Rightarrow\;x^{-3}=\frac{1}{x^3} \end{aligned}$$ So a negative exponent means reciprocal: flip it to the other floor of the fraction. It does not mean the number is negative. For positive x, x⁻³ is a perfectly ordinary positive number; the minus sign is an instruction to flip, not a sign on the value.