Exponents & Roots

Fold a piece of paper

A sheet of paper is about $0.1$ millimeters thick. Fold it in half and it’s $0.2$; again, $0.4$. Ten folds in, the stack is a full $10$ centimeters. By $23$ folds it would clear a kilometer, and around fold $42$ — if paper allowed it — the stack would reach the Moon. Nothing about one fold feels dramatic; the drama is that each step multiplies instead of adds. Doubling ten times isn’t $10 \times 2$, it’s $2 \times 2 \times \cdots \times 2$ ten times over — and math needs a notation for “multiply this by itself that many times.”

An exponent is repeated multiplication

You met the notation in F1 as the densest shorthand on the priority ladder: in $2^{4}$, the base ($2$) is what gets multiplied and the exponent ($4$) counts the copies: $2^{4} = 2 \times 2 \times 2 \times 2 = 16$. You’ve used it since, too — $60 = 2^{2} \times 3 \times 5$ in F2 leans on it. That “counts the copies” reading is the master key to this unit: every exponent rule is just what happens when you write the copies out in full.

Derive the rules — don’t memorize them

Multiply $2^{3} \cdot 2^{4}$ by unpacking both: $(2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2)$ — seven copies of $2$ in a row, so the answer is $2^{7}$. Copies stack, so same-base products add exponents. Division runs the same movie backwards: $2^{5} \div 2^{2}$ cancels two copies off the top, leaving $2^{3}$ — subtract. And a power of a power makes copies of copies: $(2^{3})^{2} = 2^{3} \cdot 2^{3} = 2^{6}$ — multiply.

Two slips account for most exponent errors, and both come from the notation looking more symmetric than it is. First, $2^{3} \cdot 5^{4}$ looks like it should combine the way $2^{3} \cdot 2^{4}$ does — but write the copies out and there’s nothing to stack: three $2$s and four $5$s share no common base, so the rule simply doesn’t apply. Second, “add or multiply?” blurs under time pressure; the copies decide instantly. Stacking rows of copies adds; copying the whole row multiplies.

It opens on the product rule with $2^{3} \cdot 2^{4}$. Before switching to each other rule, predict the exponent it will produce from the same $3$ and $4$ — the quotient rule should give $-1$ (a small number, not a negative one!), and power-of-a-power $12$.

The pattern behind zero and negative exponents

What could $2^{0}$ mean — zero copies of $2$? The gut answer is $0$, since zero copies feels like nothing. And $2^{-3}$ looks like it should be negative. Both instincts break against a pattern you can verify: step the exponent down by one and the value divides by the base each time — $2^{3}=8$, $2^{2}=4$, $2^{1}=2$. The ladder doesn’t stop there: one more step down forces $2^{0} = 1$ (divide $2$ by $2$), then $2^{-1}=\frac12$, $2^{-2}=\frac14$, $2^{-3}=\frac18$. Zero and negative exponents aren’t a new arbitrary rule — they’re the only values that keep the dividing pattern unbroken. So $a^{0}=1$ for any non-zero $a$, and $a^{-n} = \frac{1}{a^{n}}$: not negative, just small.

The calculator opens on $2^{-3}$ — check it against the ladder. Then try $7^{0}$, and $10^{-2}$: keep that last one in mind, it comes back in scientific notation.

Roots undo powers

Every operation earns an undo eventually. A square root asks “what number, squared, gives this?” — $\sqrt{49} = 7$ because $7^{2} = 49$. When the number isn’t a perfect square, you can still partially undo it using the atoms from F2: find the largest perfect-square factor hiding inside, and pull its root out front. For $\sqrt{72}$: since $72 = 36 \cdot 2$, split the root — $\sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$, an exact answer with the square part extracted. Cube roots undo cubing the same way, pulling out perfect cubes: $\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$.

Predict before simplifying: what’s the largest perfect square inside $72$? (Not $4$ — go bigger.) Then try $50$, then $48$ — and $47$, which refuses to simplify at all (its atoms, $47$, contain no squares).

Scientific notation — taming huge and tiny numbers

The Earth weighs about $5{,}970{,}000{,}000{,}000{,}000{,}000{,}000{,}000$ kilograms, and a water molecule is about $0.000000000275$ meters wide. Both numbers are almost all zeros — the only information is one short string of digits and how far from the decimal point it sits. Scientific notation stores exactly those two facts: $a \times 10^{n}$, where $a$ keeps a single non-zero digit before the point and the power of ten counts the point’s hops (the same hops you counted in F4). $5300 = 5.3 \times 10^{3}$ — point hopped $3$ left. $0.00042 = 4.2 \times 10^{-4}$ — point hopped $4$ right, and there’s your negative exponent meaning small, not negative.

Convert $5300$, then $0.00042$ and check each exponent against your hop count. Then give it the Earth’s mass — type the digits and count the zeros yourself first.

The one thing to remember

An exponent counts copies in a multiplication, and every rule falls out of writing the copies down: stacking copies adds exponents, canceling subtracts, copying the whole row multiplies — same base only. Stepping the exponent down divides by the base, which is why $a^{0} = 1$ and negative exponents mean small. Roots run the whole machine in reverse.

What an exponent means

An exponent is repeated multiplication: $2^{4} = 2 \times 2 \times 2 \times 2 = 16$. The base is what’s multiplied; the exponent is how many times.

The six exponent rules

Roots

A square root undoes squaring: $\sqrt{49} = 7$ because $7^{2} = 49$. A cube root undoes cubing: $\sqrt[3]{27} = 3$. To simplify a radical, pull out the largest perfect-square factor (largest perfect cube for a cube root): $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

Scientific notation

A compact way to write very big or very small numbers: $a \times 10^{n}$ where $a$ has one non-zero digit before the point. $5300 = 5.3 \times 10^{3}$; $\ 0.00042 = 4.2 \times 10^{-4}$. A positive exponent means a big number (point moved left); a negative exponent means a small one (point moved right).