Rationalizing Denominators

A fraction like is a perfectly valid number, but it isn’t considered a finished algebra answer — by convention, a denominator is supposed to be free of radicals. Getting there always comes down to multiplying the fraction by a cleverly disguised form of , chosen specifically to turn the denominator into a perfect power. This lesson covers every version of that multiplier you’ll need.

Radicals introduces rationalizing as part of a broader toolkit; this lesson is the focused, example-heavy version, covering single-term denominators, two-term (binomial) denominators, and higher-index roots one at a time.

Why Rationalize a Denominator?

Rationalizing a denominator means rewriting a fraction so that no radical remains on the bottom, without changing the fraction’s value. The tool for doing this without changing the value is always the same: multiply by a fraction equal to , built from whatever clears the radical.

equals exactly , so multiplying by it never changes what the fraction is worth — only moved from the denominator to the numerator, where a radical is considered acceptable.

Rationalizing a Single-Term (Monomial) Denominator

When the denominator is a single radical, multiply top and bottom by just enough of that same radical to complete a perfect power underneath the root.

For a square root, one extra copy always finishes the job, since . For a higher-index root, the multiplier needs to complete the radicand up to a full power of the index — not necessarily an identical second copy, as the next section shows.

Rationalizing a Two-Term (Binomial) Denominator with a Conjugate

When the denominator has two terms and at least one is a radical, a single multiplier can’t clear it — squaring directly would still leave a radical behind, since . The fix is multiplying by the conjugate instead: the same two terms with the sign between them flipped.

This is the difference of squares pattern, and it’s exactly why the conjugate works: squaring a square root always removes its radical, and a sum times its conjugate is defined to produce that exact pattern.

Rationalizing Denominators with a Higher-Index Root

For an index- root, one extra factor doesn’t necessarily finish clearing the radical — the multiplier needs to bring the radicand’s exponent up to a multiple of .

The original radicand was ; multiplying by brought the total exponent under the root up to , which is a perfect cube and clears the root entirely. A common mistake is multiplying by another copy of instead, which only reaches — still not a perfect cube.

Worked Example A: A Monomial Square-Root Denominator

Rationalize .

Step 1 — multiply by :

Answer:

Worked Example B: A Monomial Denominator That Needs Simplifying First

Rationalize .

Step 1 — simplify the radical in the denominator first:

Step 2 — rationalize the simplified denominator:

Answer: . Simplifying first avoided a much larger multiplier and an extra simplification step at the end.

Worked Example C: A Binomial Denominator with a Conjugate

Rationalize .

Step 1 — multiply by the conjugate of the denominator, over itself:

Step 2 — expand the denominator using the difference of squares pattern:

Step 3 — expand the numerator by distributing:

Answer:

Worked Example D: A Higher-Index Root Denominator

Rationalize .

Step 1 — rewrite as , so more factors of are needed to reach :

Step 2 — multiply out:

Answer:

Common Mistakes to Avoid

• Multiplying only the denominator by the rationalizing factor. The numerator has to be multiplied by the exact same thing, since the multiplier must equal — multiplying only one part changes the fraction’s value.
• Squaring a binomial denominator instead of using its conjugate. still contains a radical; only multiplying by the conjugate produces a difference of squares.
• Flipping the wrong sign when writing a conjugate. The conjugate of is — only the sign between the two terms changes, not the sign of either term individually.
• Using an identical copy of the radical for a higher-index root. For , multiplying by another only reaches under the root, not a perfect cube; the correct multiplier is .
• Forgetting to distribute across a binomial numerator after multiplying by a conjugate. Every term in the numerator gets multiplied by every term in the conjugate, the same as any other binomial expansion.
• Rationalizing before simplifying the original radical. Skipping the simplification step often leads to a larger, messier multiplier and an unsimplified final answer.

Where This Shows Up Later

• Complex Number Arithmetic. Dividing by a complex number uses the exact same conjugate trick, since times its conjugate also produces a real, radical-free (and imaginary-free) result.
• The quadratic formula. An answer like sometimes needs its denominator rationalized further if it appears inside a larger fraction elsewhere in a problem.
• Limits in calculus. Rationalizing a numerator, using the identical conjugate technique from this lesson, is a standard trick for evaluating a limit that would otherwise divide by zero.
• Trigonometric identities. Exact trig values like are conventionally rationalized to before being reported as a final answer.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Rationalize .

Problem 2. Rationalize .

Problem 3. Rationalize .

Problem 4. Rationalize .

Problem 5. Rationalize .

Problem 6. Rationalize .

Problem 7. Rationalize .

Problem 8. Rationalize .

Problem 9. Rationalize .

Problem 10. Rationalize .

Quick Reference

The same conjugate idea shows up again in Complex Number Arithmetic, just dividing by a complex number instead of clearing a radical. If a denominator’s radical needs simplifying before rationalizing, Simplifying Radicals covers that step, and Radicals has the full toolkit both lessons are drawn from. Practice these moves with the Rationalizing Denominators Practice Problems, or check out more of the Algebra lessons.