L7 · Factors and roots clinic

Learning goal

“I can move fluently among everything in this unit — prime factorizations, GCF and LCM problems, perfect powers, ordering — and I can rewrite a radical in simplest form (mixed radical) or as an entire radical, checking each rewrite by squaring or cubing it back.”

Mixed radical — a radical with a coefficient outside: 6√2. The radicand is left as small as possible.

Entire radical — a radical with no coefficient outside: √72. The whole value lives under one radical sign.

Simplest form — a mixed radical whose radicand has no perfect-square factor (for index 2) or perfect-cube factor (for index 3) left inside: √72 rewritten as 6√2 is in simplest form.

Worked example

Express √72 in simplest form. Step 1: Find the largest perfect-square factor of 72: 72 = 36 × 2. Step 2: The root of the perfect square comes OUTSIDE: √72 = √36 × √2 = 6√2. Step 3: Check by squaring the answer: (6√2)² = 6² × (√2)² = 36 × 2 = 72 ✓ — the rewrite is exact, not an approximation. Step 4: Cube version: ∛54 = ∛(27 × 2) = 3∛2; check: (3∛2)³ = 27 × 2 = 54 ✓ Say it plainly: you extract the ROOT of the perfect-power factor (36 -> 6), never the factor itself.

Check yourself — explain it

Why does √72 become 6√2 and not 36√2?

Paired problem — example still visible

Express √147 in simplest form. -> 7√3 (147 = 49 × 3; √49 = 7 comes out; check: (7√3)² = 49 × 3 = 147 ✓).

Faded completion — last step is yours

"√180 = √(36 × ___) = ___√5" -> 5; 6

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz

Grade 10 — Mathematics 10C › C10U04 Factors, Roots & Irrational Numbers

L7 · Factors and roots clinic