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Math · Algebra

Radicals and radical equations

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.

Practise radicals and radical equations nowSee sample questions
3
Difficulty tiers
40s
Target pace
54%
Average first-attempt score

What's covered

  • Simplifying radicals — pull out the largest perfect-square (or perfect-cube) factor from under the root.
  • Operations on radicals — combine like radicals the same way as like terms, and rationalize a denominator that contains a root.
  • Solving one-step radical equations — isolate the radical, then raise both sides to the matching power to eliminate it.
  • Checking for extraneous solutions — squaring both sides of an equation can introduce a solution that doesn't satisfy the original — always substitute back to check.
  • Radicals with variables and exponents — apply the same simplification rules when the expression under the root includes a variable raised to a power.

Where students lose marks

Forgetting to check for extraneous solutions

Every solution to a radical equation solved by squaring must be substituted back into the original equation, not just the squared version.

Adding radicals that aren't like terms

√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.

Squaring a sum instead of the whole expression

(√x + 3)² isn't x + 9 — it expands like any binomial square, with a middle term you can't skip.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

Simplify: √72

Sample 02Core

Combine: √8 + √18

Sample 03Advanced

Solve: √(x + 7) = x + 1

How to practise this

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.

Always check your answer in the original equation

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.

Then 40 seconds a question

Move to timed sessions once the check-back step is automatic. It adds time, so it needs to be budgeted, not skipped.

Fold into a mock

Combine radicals with exponents in a mock — fractional exponents and roots are two notations for the same operation.

Stop skipping the check.
Start catching the extraneous root.

  • An extraneous-solution check walked through on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your radicals accuracy and pace over time
Start practising free

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Related Math subtopics

Age, work, and mixture word problemsAlgebraic expressionsExponentsLinear equationsPolynomialsQuadratic equationsRational expressionsAll of Math