Radicals
Find the perfect-square structure first; simplify before you calculate or compare.
Radicals
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A radical becomes simpler when its radicand contains a perfect-square factor
The square root √n asks for a nonnegative number whose square is n. If n is not a perfect square, factor out its largest perfect-square factor.
Factor out perfect squares.
Combine only matching simplified radicals.
Multiply coefficients and radicands.
Isolate, square, then check.
Simplify first
Expressions such as √12 and √27 do not look alike until they become 2√3 and 3√3.
Check after squaring
Squaring can introduce a value that solves the squared equation but not the original radical equation.
Search for 4, 9, 16, 25, 36, 49, 64, 81, or 100
Example: Simplify √108.
Find the largest perfect-square factor:
Choosing the largest square usually completes the simplification in one step.
Why it works
The square part leaves the radical as an ordinary coefficient, while the nonsquare part remains inside.
Five forms you should recognize
Do not stop at 3√20 because 20 still contains the perfect square 4.
Simplify each term before deciding whether they are like radicals.
The product inside the radical becomes a perfect square.
Multiplying by √3/√3 changes the form, not the value.
Squaring gives x² − 2x − 3 = 0, with candidates x = 3 and x = −1. Check both:
x = 3: √9 = 3 ✓x = −1: √1 ≠ −1 ✗Only x = 3 is valid.
Check before you commit
- Using √(a + b) = √a + √b
- Combining radicals before simplifying
- Stopping while a perfect-square factor remains inside
- Forgetting to distribute a radical
- Leaving a simple radical in the denominator
- Keeping every root produced after squaring without checking
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Radicals FAQ
Why can’t √(a + b) be split?
Square roots distribute over multiplication under appropriate real-number conditions, not over addition. For example, √9 = 3 but √4 + √5 is not 3.
When can radicals be added?
After simplification, their radical parts must match exactly.
Why check solutions?
The squaring step is not reversible for every sign, so it can introduce extraneous candidates.
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