ACT Math: Simplify Radicals by Rationalizing Complex Denominators

When your denominator contains a sum or difference with a radical (like 3+√2), multiply both numerator and denominator by the conjugate. The conjugate of 3+√2 is 3-√2. Multiply: [(numerator)·(3-√2)]/[(3+√2)·(3-√2)]. The denominator becomes 9-2=7 (the radicals disappear by difference of squares). The conjugate method always produces a rational denominator because (a+√b)(a-√b)=a²-b.

Example: 1/(1+√3). Multiply by (1-√3)/(1-√3). Numerator: 1-√3. Denominator: 1-3=-2. Result: (1-√3)/(-2) or (√3-1)/2. Always simplify the final fraction.

Mistake 1: Forgetting to multiply the numerator by the conjugate. You must do both or the fraction's value changes. Mistake 2: Leaving your final answer with a radical in the denominator. ACT always expects a rationalized form. Mistake 3 (bonus): Using the wrong conjugate. For 2-√5, the conjugate is 2+√5, not -2+√5. Double-check your conjugate before multiplying.

Prevention: Write the conjugate clearly above your work. Use the pattern (a+√b)(a-√b)=a²-b to verify your denominator will be rational before you start.

Problem 1: 2/(√5-1). Conjugate: √5+1. Result: 2(√5+1)/4=(√5+1)/2. Problem 2: 1/(2+√3). Conjugate: 2-√3. Result: (2-√3)/(4-3)=2-√3. Problem 3: 3/(1+√2). Conjugate: 1-√2. Result: 3(1-√2)/(1-2)=3(√2-1) or 3√2-3. Redo all three without looking at the answers until you can finish each in 45 seconds.

If you made errors, identify where: was it the conjugate, the multiplication, or the final simplification? Fix that step and redo the problem immediately.

ACT Math tests rationalization because it's a core algebra skill and frequently pairs with equations and inequalities. Expect 1-2 rationalization questions per test. Students who master this technique save 2-3 minutes and avoid careless errors that cost points.

This skill feels mechanical, so drill it until you can rationalize 1/(3+√7) in under one minute without error. Once automatic, you'll move on to harder problems confident your algebra is solid.