Radicals and Simplification — SAT Math Explained
A radical is an expression involving a root (square root, cube root, etc.). Simplifying radicals means rewriting them in their simplest form by removing perfect powers from under the radical sign.
The Core Idea
Simplifying radicals relies on the product property: √(a × b) = √a × √b. You look for perfect square factors hiding inside the radical and extract them.
Key Vocabulary
The expression inside the radical sign
The positive square root — √25 = 5, not ±5
A number that is the square of an integer: 1, 4, 9, 16, 25, 36...
Eliminating radicals from the denominator of a fraction
Radicals with the same radicand — they can be added/subtracted
Simplification Process
1. Factor the radicand to find perfect square factors
2. Use √(a × b) = √a × √b to separate
3. Simplify √(perfect square) = integer
4. Write the extracted integer in front of the remaining radical
5. Verify: squaring your answer should give the original radicand
Operations With Radicals
Only like radicals (same radicand) can be combined, just like like terms
√a × √b = √(ab) — multiply radicands
√a / √b = √(a/b)
Multiply numerator and denominator by the radical in the denominator
Common Errors to Avoid
√(a + b) ≠ √a + √b (the most common radical error!)
Forgetting to simplify completely — check if remaining radicand has any perfect square factors
Not rationalizing the denominator when required
Practice: Radicals and Simplification
5 SAT-style questions. Select your answer and get an instant explanation.
Simplify: √48
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