Shorthand for multiplying over and over, the order everyone agrees on, and a first glimpse past the rationals.
Point 4 of 5 in this lesson: 5.6.4 The laws and smart shortcuts
When you add the same number to itself over and over, you reach for multiplication: 5+5+5 becomes 3×5. This lesson does the very same trick one floor up. When you multiply the same number by itself over and over, you reach for a power: 2·2·2 becomes 23. A power is just shorthand for "multiply this many copies together" — but it brings two new questions with it: what happens to the sign when the base is negative, and once powers, ×, ÷, +, and − all crowd into one expression, whose turn is first?
By the end you will write and expand powers, predict the sign of any power in your head, evaluate a mixed expression by the order everyone agrees on, find a clever shortcut through a messy calculation, and meet a number — √2 — that no fraction can ever pin down. We keep the color habit of the whole stage: positive numbers are teal, negative numbers are red, zero is slate, and a distance or magnitude is amber.
A power is a short way to write a repeated multiplication. The expression an means "multiply n copies of a together." The number being repeated, a, is called the base; the small raised number, n, is the exponent — it simply counts the copies.
Two of these powers are common enough to have nicknames. a2 is read "a squared" (it gives the area of a square with side a), and a3 is read "a cubed" (the volume of a cube). Everything in this stage is a rational number, so the base may be negative or a fraction:
(−2)3 = (−2)(−2)(−2) = −8 · (12)2 = 12·12 = 14
A power an means n copies of the base multiplied — not the base times the exponent. So 23 = 8, never 2·3 = 6. (One harmless rule for later: any nonzero base to the 0 power is 1, because there is "nothing left to multiply" — start at 1.)
Pick a base — a whole number, a negative, or a simple fraction — and an exponent from 0 to 5. Watch the repeated product spell itself out, then collapse to a value.
Because multiplying two negatives gives a positive (Lesson 5.5), the sign of a power follows a clean rhythm. A power of a positive base is always positive — nothing can flip it. A power of a negative base flips its sign with every copy, so the copies pair up:
| Base | Even exponent | Odd exponent |
|---|---|---|
| positive | positive | positive |
| negative | positive | negative |
Now the single most important piece of notation in this lesson. Parentheses decide whether the minus belongs to the base. Compare:
(−2)4 = 16 but −24 = −16. The exponent binds tighter than a bare minus sign. If you want the minus to be part of the base, you must wrap it in parentheses. Same story: (−3)2 = 9, while −32 = −9.
Set a negative base and an exponent. The even/odd light tells you the sign before you compute. Then flip the parentheses toggle OFF to see the −2⁴ trap spring.
Look at 5 − 2×32. If two people work it in different orders, they get different answers — and a number must have exactly one value. So the world agrees on an order, a line everyone stands in:
① Brackets first — anything inside ( ).
② Powers next.
③ × and ÷, working left to right.
④ + and −, working left to right.
Brackets can rewrite the whole result, because they jump to the front of the line. Put the subtraction inside parentheses and it goes first:
(5−2)×32 = 3×9 = 27
Same digits, same symbols — but 27 instead of −13, all because the brackets changed who went first.
Evaluate −4 + (−2)3÷4.
Powers first: (−2)3 = −8. → −4 + (−8)÷4.
Now ÷ before +: (−8)÷4 = −2. → −4 + (−2).
Finally add: −4 + −2 = −6.
Step through the evaluation one tier at a time. The piece about to be simplified lights up, so you can see exactly who takes their turn next.
The order of operations tells you a path that always works. But the laws of arithmetic — commutative (swap the order), associative (regroup), and distributive (split or factor) — often hand you a much shorter one. A good calculator looks for the friendly numbers first.
Hunt for pairs that round off to tens and hundreds. In (−25)×17×(−4), the two negatives make a positive and 25×4 = 100 is begging to be paired:
The distributive law lets you split an awkward factor into a friendly difference. To find 99×(−7), notice 99 = 100 − 1:
99×(−7) = (100 − 1)×(−7)
= 100×(−7) + (−1)×(−7)
= −700 + 7 = −693.
One easy "times a hundred" and a tiny correction beat a column of long multiplication.
Run distribution the other way — pull out a common factor — and sums collapse too: 37×8 + 37×2 = 37×(8+2) = 37×10 = 370. The laws don't change the answer; they change the amount of work.
Before you grind through an expression, glance for a shortcut: pair factors that round off (25 & 4, 2 & 5, 8 & 125), split a near number (99 = 100−1), or factor out what's shared. The order of operations is your safety net; the laws are your speed.
Every power can be run in reverse. "What number, squared, gives 9?" The answer is 3, because 32 = 9. That reverse question is a square root, written √9 = 3. When the inside is a perfect square — 1, 4, 9, 16, 25 — the root is a tidy whole number.
But now ask the same of 2: "what number squared is 2?" It sits between 1 (whose square is 1) and 2 (whose square is 4), so the answer is somewhere between 1 and 2. We can chase it with decimals — but watch what happens:
This is not just impatience: it can be proved that √2 is not a fraction at all — no pq of whole numbers squares to exactly 2. So √2 is a brand-new kind of number that lives between the rationals, plugging a tiny gap on the number line that fractions leave empty.
Numbers like √2 are called irrational. Together with all the rationals they form the real numbers — the complete, gap-free number line. Stage 6 (Powers, Roots & Real Numbers) builds them properly. For now, just hold the surprise: reversing a humble little squaring can carry you right out of the world of fractions.
Slide a guess and watch its square. You can get the square as close to 2 as you like — but you will never land exactly on it. That "almost, but never" is the feel of an irrational number.
A power an packs n copies of the base into a multiplication, so 23 = 8 and (½)2 = ¼. A positive base always gives a positive power; a negative base gives positive for even exponents and negative for odd — and (−2)4 = 16 is not −24 = −16, because parentheses decide whether the minus is part of the base. When operations mix, everyone waits their turn: brackets → powers → ×,÷ → +,−, each left to right. And the laws (commute, regroup, distribute) hand you shortcuts past the mess. Finally, reversing a power asks for a root; sometimes — like √2 — the answer is no fraction at all, our first peek at the real numbers.
You now have the whole toolkit for negative and rational numbers — comparing, adding, subtracting, multiplying, dividing, and raising to powers, all in the right order. Stage 6 opens the gate that √2 just cracked: powers, roots, and the full real-number line.
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 the U.S. Common Core standards for exponents and operations: 6.EE.A.1 (write and evaluate numerical expressions with whole-number exponents), 6.EE.A.2c (evaluate expressions, applying the conventional order of operations), and 7.NS.A.3 (solve problems with the four operations on rational numbers, including negatives). The closing section previews 8.EE.A.2 (square roots) and 8.NS.A.1 (rational vs. irrational numbers), so it deliberately stays intuitive — a felt taste of √2, not a proof. The single most common misconception is reading −24 as if it were (−2)4. The antidote, repeated in the figures and the quiz: parentheses decide whether the minus is part of the base; the exponent binds tighter than a bare minus, so without parentheses the minus is applied last. A close runner-up is computing an as a×n (e.g., 23 = 6) — counter it by always expanding to the repeated product the first few times.