Rational and Radical Expressions
Key rules for simplifying, combining, and solving rational and radical expressions on the PSAT.
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Questions Covered in This Set
10 cards to master
What must you do before cancelling in a rational expression?
Factor the numerator and denominator completely — you may only cancel factors, never terms.
Does (x+2)/2 simplify to x?
No. The 2 on top is a term, not a factor, so nothing cancels.
How do you find where a rational expression is undefined?
Set the original denominator equal to zero and solve; those x-values are excluded from the domain.
Simplify (x²−9)/(x²+7x+12).
(x−3)(x+3)/[(x+3)(x+4)] = (x−3)/(x+4), with x ≠ −3, −4.
How do you simplify a complex fraction like (1 + 1/x)/(1 − 1/x)?
Multiply numerator and denominator by the small denominator x, giving (x+1)/(x−1).
How do you divide rational expressions?
Multiply by the reciprocal of the second fraction, then factor and cancel.
Steps to solve a rational equation
Multiply every term by the LCD, solve, then check that no answer makes a denominator zero.
What is an extraneous solution?
An answer produced by the algebra that fails in the original equation — e.g., it zeroes a denominator or makes a square root equal a negative.
Rewrite ⁿ√(xᵐ) with a fractional exponent.
x^(m/n); so √x = x^(1/2) and √(x³) = x^(3/2).
Solve √(2x+3) = x.
Square: x²−2x−3=0 → x=3 or x=−1. Only x=3 checks, since √1=1 ≠ −1.
