Rationalizing Denominators Generator
The Rationalizing Denominators lesson covers every version of the rationalizing multiplier in detail; this generator drills the result those multipliers always produce — a whole-number denominator, checkable directly.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | A monomial square-root denominator | \(\dfrac{5}{\sqrt{7}}\), new denominator \(7\) |
| Medium | A conjugate binomial denominator | \(\dfrac{4}{3+\sqrt{5}}\), new denominator \(4\) |
| Hard | A bigger conjugate, or a cube-root denominator | \(\dfrac{2}{\sqrt[3]{9}}\), new denominator \(27\) |
Easy rationalizes a single square-root term in the denominator, where the new denominator is always exactly the original radicand.
Medium rationalizes a conjugate binomial denominator like \(a+\sqrt{b}\), where the new denominator always works out to \(a^{2}-b\).
Hard alternates between a bigger conjugate binomial and a cube-root denominator, where completing the cube always produces a denominator equal to a perfect cube.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Every answer here is a whole number — enter it exactly as computed.
- 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 |
|---|---|
| Monomial denominator | \(\dfrac{1}{\sqrt{a}} \cdot \dfrac{\sqrt{a}}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}\) |
| Conjugate denominator | \((a+\sqrt{b})(a-\sqrt{b}) = a^{2}-b\) |
| Higher-index denominator | Multiply by enough extra factors to reach a full \(n\)th power under the root |
See the Rationalizing Denominators lesson for the full derivation of each multiplier and worked examples with the numerator included.
Common mistakes to watch for
Squaring a binomial denominator instead of using its conjugate. \((a+\sqrt{b})^{2}\) still contains a radical; only multiplying by the conjugate clears it.
Flipping the wrong sign when writing a conjugate. The conjugate of \(3-\sqrt{5}\) is \(3+\sqrt{5}\) — only the sign between the two terms changes.
Using an identical copy of the radical for a higher-index root. For \(\sqrt[3]{x}\), multiplying by another \(\sqrt[3]{x}\) only reaches \(x^{2}\) under the root, not a perfect cube.
Forgetting the multiplier has to equal 1. Whatever is multiplied onto the denominator has to be multiplied onto the numerator too — this generator only checks the denominator, but a complete answer always keeps both.
Where to go next
The exact same conjugate trick reappears in the Complex Number Arithmetic Generator for dividing by a complex number. For the full technique and every worked example, see the Rationalizing Denominators lesson.