Advanced Math / Equivalent Expressions

Radical and Rational Equivalent Expressions

Radical and rational expression questions require precise use of factoring, cancellation, exponent rules, and domain restrictions. Equivalent expressions must match the original expression on its allowed domain.

Least you need to know

Factor before canceling in a rational expression.
A canceled denominator factor still creates a restriction in the original expression.
Simplify radicals by removing perfect-square factors.
Use the given domain condition when simplifying expressions such as $\sqrt{x^2}$.

Key notation

$\frac{x^2-25}{x+5}$ — A rational expression that can be simplified for $x\ne -5$
$\sqrt{72}$ — A radical expression that simplifies to $6\sqrt2$

Worked example

To simplify $\frac{x^2-25}{x+5}$ for $x\ne -5$, factor the numerator as $(x-5)(x+5)$.
Cancel the common factor $x+5$.
The expression is equivalent to $x-5$ for all allowed values of $x$.

Common mistakes

Do not cancel terms that are not common factors.
Do not ignore restrictions from the original denominator.
When simplifying radicals, take the square root of perfect-square factors.

How to recognize it

The problem asks which expression is equivalent.
A rational expression has a factorable numerator or denominator.
A radical contains a perfect-square factor.

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