An exponent counts factors — and every law of exponents is just that counting. Add when you multiply, subtract when you divide, multiply for a power of a power; what zero and negative exponents have to mean; square and cube roots; numbers that never end; and scientific notation for the very large and the very small. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
2⁵ means five 2s multiplied together. It is 32 — nowhere near 2 × 5.
Every law of exponents is that counting. Add when you multiply, subtract when you divide, multiply for a power of a power. You are never doing anything but counting factors.
Four things people get wrong
⚠️People often think…
2⁵ means 2 × 5.
The little number counts how many factors, not what to multiply by. 2⁵ = 2 · 2 · 2 · 2 · 2 = 32, and 2 × 5 = 10. Squaring is not doubling either: 6² is 36, not 12.
The exponent is a count, not a multiplier.
⚠️People often think…
You always add the exponents.
Add only when you are multiplying two powers: x³ · x⁴ = x⁷. A power of a power multiplies: (x³)⁴ = x¹². The parentheses are what tell you which one you are looking at.
Look for the parentheses before you touch anything.
⚠️People often think…
A negative exponent makes the answer negative.
A negative exponent means the reciprocal — a small positive number, never a negative one. 10⁻³ = 1/1000 = 0.001.
Negative exponent: flip it, don't negate it.
⚠️People often think…
Anything to the zero power is zero.
It is 1. Divide 5³ by 5³: same thing over itself, which is 1 — and subtracting the exponents gives 5⁰. Both are true, so 5⁰ has to be 1.
Zero power means nothing is left to multiply — so, 1.
Count the factors
1Watch one
Simplify x³ · x⁴.
x³ is three x's.
x⁴ is four x's.
Lined up, that is seven x's.
x⁷. You added because you were counting.
2Do one with me
Simplify (x³)⁴. Watch the parentheses.
Inside the parentheses there are
You have that whole group this many times:
So altogether, x to the
💬One sentence, then you move on
Why do you add the exponents when you multiply?
3Try one
What is 5⁰? Write the number.
I want a hint first
Divide 5³ by 5³. The same thing over itself is 1 — and subtracting the exponents gives 5⁰.
💬Last one — then you're done here
Why isn't a negative exponent a negative number?
Where this goes
Where this lives
Storage doubling from 64 to 128 gigabytes, interest compounding in a savings account, and any number too long to write out getting written short instead.
What this feeds
Next unit is linear equations and systems — tidy, gather, undo, check.
Write one number from your life that would be easier in scientific notation.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a long multiplication to a short one — the ladder
Repeated multiplication → a power (base, exponent) → laws for combining powers with the SAME base → zero and negative exponents from the pattern → roots undo powers → powers of ten → scientific notation → compute by handling the front numbers and the tens separately.
Count the factors. That is all an exponent ever does.
🤝 The whole topic in one idea
An exponent counts factors. Every law is just counting: multiply and you have more factors (add), divide and some cancel (subtract), a power of a power is groups of groups (multiply). If you ever forget a law, write the factors out and count.
⚠️ Traps the test loves
2⁵ is not 2 × 5. It is 2 × 2 × 2 × 2 × 2 = 32.
x³ · x⁴ = x⁷ (add). (x³)⁴ = x¹² (multiply). Look for the parentheses.
The laws need the SAME base. 2³ · 5² has no shortcut.
A negative exponent is not a negative number. 10⁻³ = 0.001.
Anything (except 0) to the zero power is 1 — not 0.
√ is not “divide by 2.” And x² = 49 has two solutions, 7 and −7.
After multiplying in scientific notation, check the front number is still between 1 and 10.
📐 The laws of exponents — the chart
Law
Rule
Example
Why
Product
add
x³ · x⁴ = x⁷
3 factors and 4 more
Quotient
subtract
x⁷ ÷ x² = x⁵
2 of the 7 cancel
Power of a power
multiply
(x³)⁴ = x¹²
4 groups of 3
Zero
equals 1
x⁰ = 1
x³ ÷ x³ = 1
Negative
one over
x⁻² = 1/x²
the pattern keeps dividing
🎯 How the test will ask
“Which expression is equivalent to…” — apply one law at a time; finish with positive exponents.
“Solve x² = 81” or “x³ = 27” — two answers for squares, one for cubes.
“Between which two integers is √50?” — bracket it with perfect squares.
Standard form ↔ scientific notation — count the places the decimal moves.
“How many times as large?” — divide: front numbers, then powers of ten.
A science context (distance, mass, population) — multiply or divide, then fix the form.
✅ Before the test, can you…
Say what 3⁴ means and find its value?
Use the product, quotient and power laws — and tell which one applies?
Explain why x⁰ = 1 and what x⁻² means?
Find square and cube roots of perfect squares and cubes, and estimate the others?
Tell a rational number from an irrational one?
Write, compare, multiply and divide numbers in scientific notation?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: decide which law applies, line numbers up by size, name the parts of a number in scientific notation, then find a classmate’s mistake and prove it. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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