Integer Exponent Generator
Knowing the exponent rules and applying them quickly under time pressure are two different skills. The Integer Exponents lesson covers where every rule comes from; this generator is where that knowledge turns into speed, with unlimited fresh problems and instant right/wrong feedback.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | Evaluate \(a^{n}\) directly | \(6^{3} = 216\) |
| Medium | Product/Quotient Rule, or \(a^{-n}\) | \(5^{4} \div 5^{2} = 25\) |
| Hard | Power-of-power, chains, sign traps | \(-5^{2} = -25\) vs \((-5)^{2} = 25\) |
Easy is single-power evaluation with small bases and exponents 2 through 4, plus an occasional \(a^{0}\) to keep the Zero Exponent Rule reflexive from the start.
Medium rotates between the Product Rule (\(a^{m} \cdot a^{n}\)), the Quotient Rule (\(a^{m} \div a^{n}\)) — which sometimes lands on a negative net exponent — and standalone negative exponents like \(a^{-n}\), where the answer is always a fraction.
Hard mixes four problem shapes: power-of-a-power (\((a^{m})^{n}\)), power of a product (\((p \times q)^{n}\)), three-term product/quotient chains, and the \(-b^{n}\) versus \((-b)^{n}\) sign distinction — the single most common exponent mistake, drilled directly instead of hoping it never comes up.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Enter each answer as a whole number or a fraction — both 1/16 and 0.0625 are accepted for the same answer.
- Check Answers scores the set and flags exactly which problems need another look
- Show Answers reveals every solution, useful for reviewing a paper attempt
- Print Worksheet outputs a clean page for offline practice or classroom handouts
The rules this generator drills
| Rule | Statement |
|---|---|
| Product Rule | \(a^{m} \cdot a^{n} = a^{m+n}\) |
| Quotient Rule | \(\dfrac{a^{m}}{a^{n}} = a^{m-n}\) |
| Power Rule | \(\left(a^{m}\right)^{n} = a^{mn}\) |
| Power of a Product | \((ab)^{n} = a^{n}b^{n}\) |
| Zero Exponent | \(a^{0} = 1\) |
| Negative Exponent | \(a^{-n} = \dfrac{1}{a^{n}}\) |
See the Integer Exponents lesson for where each of these comes from and a full worked derivation.
Common mistakes to watch for
Adding exponents instead of multiplying, or vice versa. \(a^{m} \cdot a^{n} = a^{m+n}\) (add), but \(\left(a^{m}\right)^{n} = a^{mn}\) (multiply) — mixing these up is the single most common error in this generator’s Medium and Hard problems.
Treating a negative exponent as a negative result. \(4^{-2} = \dfrac{1}{16}\), a positive fraction. The minus sign means “reciprocal,” never “negative value.”
Dropping the parentheses on a negative base. \(-5^{2} = -25\), but \((-5)^{2} = 25\) — genuinely different numbers. Check which one a problem is actually showing before you answer.
Combining exponents on different bases. The Product and Quotient Rules only combine same-base terms — there’s no shortcut for something like \(2^{3} \times 3^{2}\).
Where to go next
Once these rules feel automatic, the Rational Exponent Generator extends the same rules to fractional exponents and roots. For the full derivations and worked examples behind every rule drilled here, see the Integer Exponents lesson.