1
State the domain first
Find every value that makes a denominator zero or an even-index radicand negative, and record the resulting restrictions before simplifying rather than after.
Course · Algebra
Rational and Radical Expressions
What happens when a variable lands in a denominator or under a root. Both put values out of bounds, both create solutions that are not solutions, and both need the domain written down before the algebra starts.
a/b · c/d = ac/bd a/b ÷ c/d = a/b · d/c
x = 2 makes (x² − 4)/(x − 2) undefined, not 4
squaring both sides can INVENT a solution
degree of top vs bottom ⟹ the horizontal asymptote
What this course is about
- Subject
- Algebra
- Lessons
- 12
- Format
- Interactive
Overview
What you will be able to do
By the end you can handle an expression with a variable in a denominator or under a root without losing the values that were never allowed.
2
Simplify and combine exact expressions
Cancel only common FACTORS, build a least common denominator, reduce a complex fraction two ways, extract perfect powers, combine like radicals and rationalize the numerical denominators taught here.
3
Solve and then check
Clear denominators or square both sides, solve, and test every candidate against the original — because both moves can create solutions that are not ones.
4
Read domains and graphs
Locate rational-function asymptotes and holes from factored form, then solve an even-root inequality and use its boundaries and range to sketch a radical function.
Lessons
12 lessons
Rational expressions, equations and graphs come first; radicals follow, from simplification to radical functions.
Rational Expressions and Their DomainsFactor the original denominator and state every excluded real value before simplifying.Not marked
Simplifying Rational ExpressionsFactor both levels, cancel only shared factors, and retain every original exclusion.Not marked
Multiplying and Dividing Rational ExpressionsRewrite division as a product, cancel factors, and account for all three sources of exclusions.Not marked
Adding and Subtracting Rational ExpressionsBuild the least common denominator from factors and combine every numerator term with its sign intact.Not marked
Complex FractionsClear a complex fraction by two methods and preserve restrictions from every level.Not marked
Solving Rational EquationsClear denominators, solve the resulting linear or factorable quadratic equation, and check every candidate in the original.Not marked
Graphs and AsymptotesFactor a rational function and classify its holes, asymptotes, zeros and end behaviour without plotting first.Not marked
Simplifying Radical ExpressionsDecide the real domain and simplify an nth-root expression, using absolute value when an even root requires it.Not marked
Adding, Subtracting and Multiplying RadicalsSimplify first, combine only like radicals, and distribute products without losing cross terms.Not marked
Rationalizing DenominatorsSimplify a numerical radical denominator, choose the completing root or conjugate, and reduce the equal result.Not marked
Solving Radical EquationsIsolate one radical, raise both sides, solve, and reject every candidate that fails in the original equation.Not marked
Radical Functions and Their GraphsSolve the root condition for the domain, locate every boundary and range limit, and sketch from exact points.Not marked
Recommended background
Factoring, domain inequalities, and function notation.
Practice suggestions
Factor everything before you do anything else. Every technique in the first half is stated in terms of factors, and none of them applies to a sum.
Write the excluded values down at the top of the page, before you start. They are not recoverable from your simplified answer — that is the whole point of “Rational Expressions and Their Domains”.
After each complete example, cover its answer and do the faded rehearsal before the quiz. The first strategic move is supplied; the remaining algebra, restrictions and original-expression check are yours. In “Solving Rational Equations” and “Solving Radical Equations” that check is part of the method, because a legal step can create an extraneous candidate.
What it does not cover
Partial fractions, which are a technique for integration and belong to calculus.
Oblique and curvilinear asymptotes beyond a brief mention where long division makes one visible.
Radicals of complex numbers, and roots of negative numbers generally. Quadratics and Complex Numbers introduces i; here sqrt(-4) is outside the domain and is treated as such.