Numbers and Expressions
Numbers and Expressions tests fluency with real numbers, powers and roots, polynomial manipulation, factoring, and rational expressions. EJU Course 2 items often combine these ideas rather than ask for a single mechanical step.
Core rules
- For real a, sqrt(a^2)=|a|, not automatically a. Rationalize denominators by multiplying by the conjugate.
- Know factoring patterns: a^2-b^2, a^3±b^3, and grouping. A rational expression is defined only where its denominator is nonzero.
- A polynomial is factored completely over the reals into linear and irreducible quadratic factors; Course 2 usually expects integer or simple radical coefficients.
- When solving equations with radicals or denominators, state domain restrictions first and check candidate solutions.
Worked example
Simplify (sqrt5+1)/(sqrt5-1). Multiply numerator and denominator by the conjugate sqrt5+1: denominator becomes 5-1=4; numerator becomes (sqrt5+1)^2=5+2sqrt5+1=6+2sqrt5. Thus the expression equals (6+2sqrt5)/4=(3+sqrt5)/2. The conjugate step removes the radical from the denominator without changing the value.
Traps
Do not cancel terms across a sum, such as (x+1)/x to 1. Do not write sqrt(a)sqrt(b)=sqrt(ab) when a,b are negative in real arithmetic. Check whether a simplification changes the domain; cancelling x-x in a denominator may create an excluded point. In EJU-style multi-step problems, preserve exact forms until the final answer.