One cell that doubles every hour: 2×2×2 is tiresome to write, so we write it once and count — 23 = 8.
A single cell splits in two every hour. After one hour there are 2. After two hours, 2×2 = 4. After three, 2×2×2 = 8. The same number — 2 — keeps showing up, multiplied by itself over and over. Writing out that long chain quickly becomes silly, so mathematicians invented a shorthand: write the repeated number once and put a small count above it. That shorthand is a power, and learning to read it, write it, and evaluate it is the whole job of this lesson.
One steady color habit runs through every figure and every line of math here. The base — the number being multiplied — is teal. The exponent — the count sitting up top — is amber. The power's value — the result you get after multiplying it all out — is purple. And a negative sign is red, because in this lesson signs are something to watch like a hawk.
This lesson rests on what you already own from Stage 5: how to multiply signed numbers (negative times negative is positive) and how to multiply rational numbers (fractions). A power is just repeated multiplication, so every Stage 5 rule rides along inside it.
6.1.1 Multiplying the same number over and over
Imagine a single bacterial cell in a warm dish. Every hour it divides, so the count doubles. Start with 2 cells. Wait an hour and you have 2×2. Wait another, 2×2×2. Nothing fancy is happening — you are just multiplying by 2 again, and again, and again. The same number, over and over.
Now picture writing this for a full day. After ten hours you'd be staring at 2×2×2×2×2×2×2×2×2×2. Count those twos — did you get ten? It's hard to be sure, and that's exactly the problem. A long line of identical factors is tiresome to write and easy to miscount. Whenever notation is this clumsy, mathematicians shorten it.
Here is the shorthand. Instead of writing 2 ten times, write it once and record how many copies with a small raised number: 210. The big 2 says which number is being multiplied; the little 10 says how many of them. That compact object is a power. So 2×2×2 becomes 23, and once you multiply it out, 23 = 8.
Each hour doubles the count. Three doublings is 2×2×2 — written short, 23 = 8.
Key idea
A power is repeated multiplication written compactly: the base is the number multiplied, and the exponent counts how many copies. 25 means 2×2×2×2×2 — five factors of 2.
🎮 Try itExpand a power, factor by factor
Step the exponent from 1 to 8 and watch the chain of 2s grow — then see the same thing written short as a power.
Exponent n3
6.1.2 Base, exponent, and power
Every power has three parts, and each has a name worth knowing. In the symbol an, the lower number a is the base — the thing being multiplied. The little raised number n is the exponent — the count of how many bases are multiplied together. And the number you get after doing all that multiplying is the power, the value of the expression. Read an aloud as "a multiplied by itself n times."
English gives two of these a special spoken name. A second power, like 22, is read "twosquared" — because 22 is the area of a square with side 2 (you'll explore that in 6.2). A third power, like 23, is read "twocubed" — the volume of a cube with edge 2. From the fourth power on, we just say "to the …th": 54 is "five to the fourth power."
The anatomy of a power: base (what), exponent (how many), power (the value).
Watch out
23 is not2×3. The exponent is a count, not a factor. 23 = 2×2×2 = 8, while 2×3 = 6. Mixing these up is the single most common slip with powers — when in doubt, write out the factors.
Worked example
Read and evaluate each power.
• 32 = "threesquared" = 3×3 = 9.
• 103 = "tencubed" = 10×10×10 = 1000.
• 54 = "five to the fourth" = 5×5×5×5 = 625.
🎮 Try itThe power labeler
Pick a base and an exponent. The machine names the power aloud, expands it, and multiplies it out.
Base a2
Exponent n3
6.1.3 Powers of negatives and fractions
A base doesn't have to be a friendly counting number — it can be negative or a fraction. The meaning never changes: a power is still "multiply the base by itself that many times." What changes is that now your Stage 5 rules for signed and rational multiplication do real work inside the chain.
Take (−2)3. The base is the whole thing inside the parentheses, −2. So (−2)3 = (−2)×(−2)×(−2). Multiply step by step, carrying the sign: (−2)×(−2) = +4 (negative times negative), and then +4×(−2) = −8. The negative sign acts like a switch that flips at every step.
A fraction base is just as honest — you raise the top and the bottom separately. (23)2 = 23×23 = 2×23×3 = 49. The numerator gets squared, the denominator gets squared — that's all.
Left: the sign of (−2)n flips at every step. Right: a fraction raises its top and bottom separately, (23)2 = 49.
Parentheses change everything
(−2)2 means the base is −2: (−2)×(−2) = +4. But −22 means "the opposite of 22" — the base is just 2, the minus waits outside — so it equals −(2×2) = −4. Same digits, opposite answers. The parentheses decide whether the sign is part of the base. (More on this trap in 6.1.5.)
🎮 Try itThe negative / fraction base machine
Choose a tricky base, step the exponent, and watch each multiplication and the running sign or fraction build up.
Base
Exponent n3
6.1.4 The sign pattern of powers
Once you've multiplied a few negative bases, a clean pattern jumps out. Watch the signs of the powers of −2: (−2)1 = −2, (−2)2 = +4, (−2)3 = −8, (−2)4 = +16. Negative, positive, negative, positive — the sign flips back and forth like a rotating duty roster.
The reason is simple once you see it. Multiplying two negatives makes a positive, so the negatives pair up and cancel. With an even exponent every negative finds a partner — all the minus signs cancel in pairs — and the result comes out positive. With an odd exponent one lonely negative is left over with no partner, and that leftover drags the answer negative.
(−a)n
n even
n odd
sign of result
+ positive
− negative
why
negatives pair up
one negative left over
Negatives cancel in pairs. Even exponent → all paired → positive. Odd exponent → one left over → negative.
Key idea — the even/odd rule
For a negative base, the sign of the power depends only on whether the exponent is even or odd: even → positive, odd → negative. A positive base is always positive, no matter the exponent.
🎮 Try itThe even / odd sign roster
Step the exponent for the base −2 and watch the duty roster light up — even rows positive, odd rows negative.
Exponent n3
6.1.5 Order of operations with powers
When a power sits inside a longer expression, it has a place in line. The order is: powers first, then multiplication and division, then addition and subtraction — and parentheses cut the line, doing whatever is inside them before anything else. Powers go early because a power is itself a bundle of multiplications, and we resolve that bundle before mixing it with the rest.
Try 3 + 2×52. Powers first: 52 = 25. Then the multiplication: 2×25 = 50. Then the addition: 3 + 50 = 53. If you had carelessly added 3 + 2 first you'd get a wrong 125 — order matters.
Now the headline trap of this whole lesson. Compare (−2)4 with −24. They look almost identical, yet:
(−2)4 = 16 but −24 = −16
Why? In (−2)4 the parentheses make −2 the base, so all four factors are −2 and the four negatives pair off to give +16. In −24 there are no parentheses, so the exponent binds only to the 2 right beneath it — the leading minus stays outside and waits. You compute 24 = 16 first, then apply the minus: −16. The exponent grabs tighter than the leading minus.
Parentheses decide the base. With them, (−2)4 = 16. Without them, the exponent binds tighter than the minus, so −24 = −16.
The #1 trap
A leading minus with no parentheses is not part of the base. −32 = −9 (do 32 = 9, then negate), while (−3)2 = +9. If you want the negative number itself to be the base, you must wrap it in parentheses.
🎮 Try itOrder-of-operations stepper
Switch between the two famous traps and the 3 + 2×5² example, then step through the evaluation one stage at a time.
Expression
★ The big ideas, in one breath
A power is repeated multiplication written once and counted: the base says what you multiply, the exponent says how many times, the power is the value — so 23 = 2×2×2 = 8, and never 2×3. Negative and fraction bases follow your Stage 5 rules step by step; a negative base turns positive for even exponents and negative for odd ones (negatives cancel in pairs). And in a longer expression, do powers first — but remember that parentheses decide the base, so (−2)4 = 16 while −24 = −16.
Coming up next — 6.2
The very first power, the square, has a question hiding behind it: if 52 = 25, what number squared gives 25? That backward question is the square root — the star of 6.2 · Squares & Square Roots.
✎ Exercises 6.1
Work each one out first, then open the answer to check your thinking.
Write 6×6×6×6 as a power.
Show answer
Four factors of 6, so it is 64. The base is the repeated number; the exponent counts the factors.
Read 72 and 43 aloud, then evaluate each.
Show answer
72 is "sevensquared" = 7×7 = 49. 43 is "fourcubed" = 4×4×4 = 64.
Is 23 equal to 2×3? Explain.
Show answer
No. 23 = 2×2×2 = 8, but 2×3 = 6. The exponent counts the factors; it is not a factor itself.
Evaluate 104.
Show answer
10×10×10×10 = 10000. (A handy fact: 10n is a 1 followed by n zeros.)
Without computing the full value, decide the sign of (−5)7.
Show answer
The exponent 7 is odd, so a negative base gives a negative result. (One negative is left over after pairing.) The sign is −.
Find the value of −24 and of (−2)4. Are they the same?
Show answer
No. −24: the exponent binds only to 2, so 24 = 16, then the minus gives −16. (−2)4: the base is −2, four negatives pair off, giving +16. The parentheses change the answer.
Evaluate 3 + 2×52 using the order of operations.
Show answer
Powers first: 52 = 25. Then multiply: 2×25 = 50. Then add: 3 + 50 = 53.
Evaluate (−2)3 + 3×(−1)4.
Show answer
Powers first: (−2)3 = −8 (odd → negative) and (−1)4 = +1 (even → positive). Then multiply: 3×1 = 3. Then add: −8 + 3 = −5.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson serves Common Core 6.EE.A.1 — "write and evaluate numerical expressions involving whole-number exponents." It also leans on Stage 5's rules for multiplying signed and rational numbers (the operations behind 6.NS.C and 7.NS.A.2), which power notation simply repeats. The #1 misconception is reading 23 as 2×3 = 6 instead of 2×2×2 = 8 — treating the exponent as a factor rather than a count. The antidote is to always expand the chain in full before evaluating, and to say it aloud as "two multiplied by itself three times." The close runner-up is the sign trap −24 = −16 versus (−2)4 = 16; cure it by asking, every time, "is the negative sign inside the parentheses, and therefore part of the base?"
eastmath.com · Stage 6 · 6.1 From Repeated Multiplication to Powers · Intuition before notation
eastmath.com · 6.1 From Repeated Multiplication to Powers · 6.1.2 Base, exponent, and power