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Rationalizing a denominator removes square roots from the bottom of a fraction, which is the conventional way to express such fractions in simplest form. This tool rationalizes denominators of the form a/√b, or a/(b + √c) and a/(b − √c) using the conjugate.
How it works
For a/√b, multiplying top and bottom by √b clears the root from the denominator. For a/(b ± √c), multiplying top and bottom by the conjugate (b ∓ √c) uses the difference-of-squares identity (b + √c)(b − √c) = b² − c to eliminate the root from the denominator.
- Enter form of the expression.
- Enter a (numerator).
- Enter b.
- Enter c (only for a / (b ± √c)).
- Click Calculate to see your results.
Examples
A binomial denominator
1/(2 + √3), multiplied by the conjugate (2 − √3)/(2 − √3), gives (2 − √3)/(4 − 3) = 2 − √3.
Who should use it
- Algebra and precalculus coursework on radical expressions.
- Simplifying radical expressions before further algebraic manipulation.
Industry applications
- Mathematics education (algebra)
Advantages
- Handles both the single-term (a/√b) and binomial (a/(b ± √c)) cases.
- Shows the conjugate multiplication and resulting simplification explicitly.
Limitations
- Doesn't handle denominators with cube roots or higher-order radicals, only square roots.
Common mistakes to avoid
- Multiplying by the same expression instead of its conjugate for a binomial denominator, which does not eliminate the square root.
Best practices
- Always multiply both the numerator and denominator by exactly the same expression (the conjugate), so the value of the overall fraction is unchanged.
Tips
- For a single square root term in the denominator, you only need to multiply by that same root (not a full conjugate) — the conjugate technique is specifically for binomial denominators.