Negative Exponent Generator
The Negative Exponents lesson covers exactly why a negative exponent means reciprocal, never a negative value — this generator is where that single rule turns into speed, with unlimited fresh problems across numbers, products, quotients, and variable expressions.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | Evaluate \(a^{-n}\) directly, including a negative base | \(3^{-2} = \dfrac{1}{9}\) |
| Medium | Negative exponent on a quotient or product | \(\left(\dfrac{2}{5}\right)^{-2} = \dfrac{25}{4}\) |
| Hard | Chained negative exponents, or a coefficient-variable expression at a chosen \(x\) | \(3^{-1} \cdot 3^{-2} = \dfrac{1}{27}\) |
Easy evaluates \(a^{-n}\) directly with small whole-number bases, occasionally with a negative base to keep the even/odd sign rule reflexive from the start.
Medium applies the negative exponent rule to a quotient (flip the fraction, drop the minus sign) or a product, so the reciprocal is taken of more than a single number.
Hard mixes two problem shapes: chaining two negative exponents on the same base with the Product Rule, and evaluating a coefficient-variable expression like \(5x^{-2}\) at a specific \(x\), which tests that the coefficient never moves.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Enter each answer as a fraction or its decimal equivalent — both 1/9 and 0.111 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 |
|---|---|
| Negative Exponent | \(a^{-n} = \dfrac{1}{a^{n}}\) |
| Negative Exponent of a Product | \((ab)^{-n} = \dfrac{1}{a^{n}b^{n}}\) |
| Negative Exponent of a Quotient | \(\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}\) |
See the Negative Exponents lesson for the full reasoning behind this rule and why it never produces a negative result on its own.
Common mistakes to watch for
Treating the negative exponent as a negative number. \(3^{-2} = \dfrac{1}{9}\), a small positive fraction, never \(-9\).
Moving a coefficient that has no exponent attached. In \(5x^{-2}\), the \(5\) stays exactly where it is — only \(x^{-2}\) becomes a fraction.
Forgetting a negative base’s sign still applies. \((-3)^{-3} = -\dfrac{1}{27}\); the negative base carries through exactly as it would with a positive exponent.
Distributing a negative exponent over a sum. The negative exponent rule only ever applies to a single factor, a product, or a quotient — never to two terms being added or subtracted.
Where to go next
Once the reciprocal rule feels automatic on its own, the Exponent Rules Generator combines it with the product, quotient, and power rules in a single problem. For the full derivation and every sign-related mistake this rule invites, see the Negative Exponents lesson.