When you multiply the same number again and again, write it once and count.
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.
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.
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.
Step the exponent from 1 to 8 and watch the chain of 2s grow — then see the same thing written short as a 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 "two squared" — 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 "two cubed" — 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."
23 is not 2×3. 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.
Read and evaluate each power.
• 32 = "three squared" = 3×3 = 9.
• 103 = "ten cubed" = 10×10×10 = 1000.
• 54 = "five to the fourth" = 5×5×5×5 = 625.
Pick a base and an exponent. The machine names the power aloud, expands it, and multiplies it out.
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.
(−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.)
Choose a tricky base, step the exponent, and watch each multiplication and the running sign or fraction build up.
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 |
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.
Step the exponent for the base −2 and watch the duty roster light up — even rows positive, odd rows negative.
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.
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.
Switch between the two famous traps and the 3 + 2×5² example, then step through the evaluation one stage at a time.
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.
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.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
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?"