Simplify expressions under a root and solve equations that contain one — the part everyone forgets is checking for extraneous solutions afterward. Every question comes with a written explanation of where a solution needs to be checked, not just computed.
Every solution to a radical equation solved by squaring must be substituted back into the original equation, not just the squared version.
√8 and √18 both simplify to a multiple of √2, but they look unrelated until simplified. Combining what's under two different roots directly — √a + √b as √(a+b) — is never valid.
(√x + 3)² isn't x + 9 — it expands like any binomial square, with a middle term you can't skip.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Simplify: √72
A — 6√2. 72 factors as 36 × 2, and 36 is a perfect square, so √72 = √36 · √2 = 6√2. 2√18 (B) is mathematically equal but not fully simplified — 18 still has a perfect-square factor (9) hiding inside it, so it isn't the simplest form the question is asking for.
Combine: √8 + √18
A — 5√2. Simplify each radical before combining: √8 = 2√2 and √18 = 3√2 — once simplified, they're like radicals and add directly to 5√2. √26 (B) comes from adding what's under the root (8 + 18 = 26) before simplifying, which isn't a valid operation — radicals don't combine that way, the same as √a + √b ≠ √(a + b) in general.
Solve: √(x + 7) = x + 1
A — x = 2. Squaring both sides gives x + 7 = (x + 1)², which simplifies to x² + x − 6 = 0, factoring to (x + 3)(x − 2) = 0 — two candidate solutions, x = −3 and x = 2. Substituting back into the original equation, x = −3 gives √4 = 2 on the left but −2 on the right, so it's extraneous and must be rejected. Only x = 2 checks out. x = 2 or x = −3 (C) is the answer you get by stopping at the algebra and skipping the check that a radical equation always requires.
Radical equations are the one Algebra subtopic where the solving isn't the hard part — the check afterward is what separates a right answer from a wrong one that looks right.
Make substituting back a non-negotiable last step on every radical equation, even when you're confident. Untimed practice is where that habit gets built before it costs a mark.
Move to timed sessions once the check-back step is automatic. It adds time, so it needs to be budgeted, not skipped.
Combine radicals with exponents in a mock — fractional exponents and roots are two notations for the same operation.
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