Radical Generator
Simplifying, multiplying, and rationalizing radicals are procedural skills — the Radicals lesson covers the reasoning behind each one, and this generator is where that reasoning 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 perfect square or cube root | \(\sqrt{81} = 9\) |
| Medium | Multiply two radicals to a perfect square | \(\sqrt{2} \cdot \sqrt{8} = 4\) |
| Hard | Conjugates, larger products, decimal approximations | \((4+\sqrt{12})(4-\sqrt{12}) = 4\) |
Easy evaluates a single perfect square or perfect cube root, so the answer is always a clean whole number.
Medium multiplies two radicals whose product lands on a perfect square, applying the Product Rule for Radicals from the lesson: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\), then simplifying.
Hard rotates between three shapes: a conjugate product like \((a+\sqrt{b})(a-\sqrt{b})\), which always eliminates the radical entirely; a larger radical product; and a genuine decimal approximation of a non-perfect root, entered to two decimal places.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Whole-number answers go in as-is; the decimal-approximation problems at Hard accept a small margin around the rounded value.
- 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 for Radicals | \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) |
| Power of a Radical | \(\left(\sqrt[n]{a}\right)^{n} = a\) |
| Conjugate Product | \((a+\sqrt{b})(a-\sqrt{b}) = a^{2}-b\) |
See the Radicals lesson for the full derivation of each rule, plus simplifying, combining, and rationalizing techniques this generator doesn’t cover directly.
Common mistakes to watch for
Forgetting to multiply the coefficients too. \((4\sqrt{7})(3\sqrt{7})\) isn’t \(\sqrt{7}\) times something small — the \(4\) and \(3\) multiply together first, giving \(12 \cdot 7 = 84\).
Getting a conjugate’s sign backward. The conjugate of \(a-\sqrt{b}\) is \(a+\sqrt{b}\) — only the middle sign flips.
Rounding a decimal approximation too aggressively. \(\sqrt{54} \approx 7.35\), not \(7.3\) or \(7\) — keep two decimal places for the Hard-tier approximation problems.
Assuming a non-perfect square has no real answer. Every radicand here is nonnegative; a decimal approximation always exists even when the exact value is irrational.
Where to go next
Once these feel automatic, the Rational Exponent Generator practices the exact same ideas written as fractional exponents instead of radical signs. For the full reasoning behind every rule drilled here, see the Radicals lesson.