Mathematics · Algebra

Radical Operations and Rationalizing Denominators

Reference entry · last updated September 20, 2026

A radical is an expression of the form \( \sqrt[n]{a} \), read as the \(n\)-th root of \(a\). The principal \(n\)-th root is the nonnegative root when \(n\) is even, and the real root when \(n\) is odd. Radicals are exponents in another notation.

1. First principles

The symbol \( \sqrt[n]{a} \) denotes the number whose \(n\)-th power is \(a\). Equivalently, \( \sqrt[n]{a} = a^{1/n} \). For even \(n\), a real root exists only when \( a \ge 0 \). For odd \(n\), roots are defined for all real \(a\), including negatives [1, 2].

2. Product and quotient rules

\( \sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}, \qquad \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \)

The product rule requires both radicands to be real. For even \(n\) this means both nonnegative. The rules follow directly from the corresponding exponent rules.

3. Simplifying radicals

A radical is simplified when no perfect \(n\)-th power remains as a factor of the radicand and no radical appears in a denominator. Extract perfect powers using the product rule. Example: \( \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \).

4. Adding and multiplying

Add or subtract only like radicals, which have the same index and radicand: \( 3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} \).
Unlike radicals cannot be combined: \( \sqrt{2} + \sqrt{3} \) stays as is.
Multiply using the product rule and simplify the result.

5. Rationalizing denominators

Rationalizing removes radicals from a denominator. For a single square-root denominator, multiply the numerator and denominator by that root:

\( \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)

For a binomial denominator containing a radical, multiply by the conjugate, \(a - b\) for \(a + b\), because \( (a+b)(a-b) = a^2 - b^2 \) clears the radical:

\( \frac{1}{\sqrt{2} + 1} = \frac{\sqrt{2} - 1}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{\sqrt{2} - 1}{1} \)

6. Radical equations and extraneous roots

To solve an equation with a square root, isolate the radical, square both sides, and solve. Squaring can introduce extraneous solutions, values that satisfy the squared equation but not the original. Check every solution in the original equation and discard those that fail.

References

^OpenStax, College Algebra 2e, Rice University (Radicals and rational exponents).
^Michael Sullivan, Algebra and Trigonometry, 11th ed., Pearson, 2019.