Radicals Practice Problems
Simplifying, multiplying, and rationalizing radicals are procedural skills — the Radicals lesson covers the reasoning behind each one, and this practice set is where that reasoning turns into speed, with unlimited fresh problems and instant right/wrong feedback.
Radical problems with answers
The generator drills radical arithmetic. Radical questions on tests also ask you to simplify, rationalize and solve, so here is one worked example of each type, from easy to hard.
1. Simplify a square root.
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2. Simplify a cube root.
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3. Multiply radicals.
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4. Rationalize a denominator.
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Multiply top and bottom by
5. Rationalize with a conjugate.
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Multiply by
6. Solve a radical equation (hard).
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Square both sides:
Each type has its own practice set: simplifying radicals, rationalizing denominators and radical equations.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | Evaluate a perfect square or cube root | |
| Medium | Multiply two radicals to a perfect square | |
| Hard | Conjugates, larger products, decimal approximations |
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:
Hard rotates between three shapes: a conjugate product like
Using the practice problems
Select a difficulty and a question count to get an instant, freshly generated set. 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 full set instantly and marks the ones worth revisiting
- Show Answers displays every solution, useful for reviewing a worksheet by hand
- Print Worksheet produces a clean, printable page, ideal for homework or in-class handouts
The rules these practice problems drill
| Rule | Statement |
|---|---|
| Product Rule for Radicals | |
| Power of a Radical | |
| Conjugate Product |
See the Radicals lesson for the full derivation of each rule, plus simplifying, combining, and rationalizing techniques this practice set doesn’t cover directly.
Common mistakes to watch for
Forgetting to multiply the coefficients too.
Getting a conjugate’s sign backward. The conjugate of
Rounding a decimal approximation too aggressively.
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 Practice Problems 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.