Rationalizing Denominators Practice Problems

Every version of the rationalizing multiplier gets a full explanation in the Rationalizing Denominators lesson. What those multipliers always produce — a whole-number denominator — is exactly what this practice set checks.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	A monomial square-root denominator	, new denominator
Medium	A conjugate binomial denominator	, new denominator
Hard	A bigger conjugate, or a cube-root denominator	, new denominator

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 , where the new denominator always works out to .

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.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Rationalize .

Multiply top and bottom by : . The new denominator is .

Medium. Rationalize .

Multiply by the conjugate : the denominator becomes , so the result is .

Hard. Rationalize .

The conjugate is , and . So the result is .

Using the practice problems

Choose a difficulty tier and a count, and a fresh problem set loads without delay. Every answer here is a whole number — enter it exactly as computed.

Check Answers grades everything in one click and shows which problems need another try
Show Answers displays all the answers immediately, useful for checking work solved by hand
Print Worksheet generates a tidy print version, ideal for classroom use or offline review

The rules these practice problems drill

Rule	Statement
Monomial denominator
Conjugate denominator
Higher-index denominator	Multiply by enough extra factors to reach a full 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. still contains a radical; only multiplying by the conjugate clears it.

Flipping the wrong sign when writing a conjugate. The conjugate of is — only the sign between the two terms changes.

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.

Forgetting the multiplier has to equal 1. Whatever is multiplied onto the denominator has to be multiplied onto the numerator too — this practice set 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 Practice Problems for dividing by a complex number. For the full technique and every worked example, see the Rationalizing Denominators lesson.

Frequently Asked Questions

Why does every problem ask for the new denominator instead of the full rationalized fraction?+

A fully rationalized fraction like (2-√3)/1 is an expression with a radical still in the numerator, which can't be automatically checked as a single number. Rationalizing always turns the denominator into a whole number, though, so asking for that one piece is a completely reliable check.

How do I find the new denominator for a monomial radical, like 5/√7?+

Multiplying by √7/√7 always turns the denominator into exactly the radicand — 7 in this example — since √a times √a is a. No other calculation is needed for a simple square-root denominator.

How do I find the new denominator for a conjugate binomial, like 1/(3+√5)?+

Multiplying by the conjugate always produces a difference of squares in the denominator: (a+√b)(a-√b) = a² − b. For 3+√5 that's 3² − 5 = 4, without needing to expand the numerator at all.

Why does a cube-root denominator sometimes need more than one extra factor?+

A cube root needs its radicand raised to a full multiple of 3 to clear completely, so if the radicand is already r² under the root, only one more factor of r is needed; if it's just r, two more factors are needed. Either way, the resulting denominator is always r cubed.

Do I need Simplifying Radicals before this practice set?+

It helps, since a denominator is usually simplified first in a full problem, but this practice set's denominators are already in a form ready to rationalize directly, so it can be practiced on its own.

Do I have to pay after a certain number of sets?+

No limit. Each set is created fresh on demand, free of charge and login-free.

1/√xRationalizing Denominators Practice Problems