Topic A: Simplifying Radicals
Lessons 1–4 · Simplifying, adding, and multiplying radicals.
01–02 — Lessons 1–2
Simplifying Square Roots
- √(ab) = √a · √b
- Factor out perfect squares: √72 = √(36·2) = 6√2
- Like radicals: same radicand. 3√5 + 2√5 = 5√5.
- Multiplying: √a · √b = √(ab)
- Rationalizing: eliminate radicals from denominators by multiplying by the conjugate.
√48 = √(16·3) = 4√3 · 2√3 + 5√3 = 7√3 · √6 · √2 = √12 = 2√3
Practice — Lessons 1–2
Q1 · Short Answer
Simplify √75.
Q2 · Multiple Choice
Which is equal to √50?
03–04 — Lessons 3–4
Rational Exponents
- aˢ/ᵗ = (ᵗ√a)ˢ = ᵗ√(aˢ)
- a^(1/2) = √a · a^(1/3) = ∛a
- Combine exponent rules with rational exponents.
- Example: 8^(2/3) = (∛8)² = 2² = 4
27^(2/3) = (∛27)² = 3² = 9 · 4^(3/2) = (√4)³ = 2³ = 8
Practice — Lessons 3–4
Q3 · Short Answer
Evaluate 16^(3/4).
Q4 · Multiple Choice
Which is equal to x^(1/3)?
Topic B: Solving Radical Equations
Lessons 5–7 · Isolating radicals and checking for extraneous solutions.
05–07 — Lessons 5–7
Solving Radical Equations
- Isolate the radical on one side, then square (or cube, etc.) both sides.
- Solve the resulting equation.
- Check ALL solutions — squaring can introduce extraneous solutions.
- Extraneous solution: satisfies the squared equation but not the original.
- For √(expression): the result must be ≥ 0.
√(x+4) = 6 → square: x+4 = 36 → x = 32. Check: √36 = 6 ✓
Practice — Lessons 5–7
Q5 · Short Answer
Solve √(3x − 2) = 4.
Q6 · Multiple Choice
After squaring both sides of a radical equation, you get x = 9 and x = 1. How do you decide which are valid?