Advanced Math / Equivalent Expressions
Equivalent Expressions with Radicals and Rational Expressions
Equivalent-expression questions often require factoring before canceling, applying exponent rules, or simplifying radicals. Restrictions such as $x\ne4$ matter because canceling a factor does not make the original expression defined at the canceled value.
Least you need to know
- Factor polynomials completely before canceling common factors in a rational expression.
- A canceled factor still creates a restriction from the original denominator.
- Simplify radicals by factoring out the greatest perfect-square factor.
- When simplifying $\sqrt{x^2}$, use the given domain condition to decide whether it equals $x$ or $|x|$.
Key notation
- $\frac{x^2-16}{x-4}$, $x\ne4$ — An expression equivalent to $x+4$ for all allowed values of $x$
- $\sqrt{50}$ — A radical expression that simplifies to $5\sqrt{2}$
Worked example
- To simplify $\frac{x^2+7x+10}{x+5}$ for $x\ne -5$, factor the numerator: $x^2+7x+10=(x+5)(x+2)$.
- Cancel the common factor $x+5$.
- The expression is equivalent to $x+2$ for $x\ne -5$.
Common mistakes
- Do not cancel terms across addition; cancel only common factors.
- Do not drop domain restrictions after simplifying a rational expression.
- For radical simplification, only perfect-square factors can be taken outside the radical.
How to recognize it
- The problem asks which expression is equivalent.
- A rational expression has a factorable numerator or denominator.
- A radical has a coefficient or variable expression with a perfect-square factor.
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