Week 3 · Operations on radicals: like radicals, inverses, conjugates and the laws of square and cube roots · 3 lessons
3
Laws of square and cube roots: products, quotients and simplest form
Mathematics Prep 2 Term 1, Unit 1, Lessons 3–4
GoalUse √a × √b = √(ab) and √a ÷ √b = √(a ÷ b), write roots in simplest form and rationalise simple quotients.
New words
product law · √a × √b = √(ab) for a, b ≥ 0quotient law · √a ÷ √b = √(a ÷ b) for a ≥ 0, b > 0simplest form · no perfect-square (or cube) factor left under the root and no root in a denominatorcube root ∛ · the number whose cube is the given number: ∛8 = 2
Explanation
Why is √4 × √9 equal to √36? Because 2 × 3 = 6 and √36 = 6. In general √a × √b = √(ab) and √a ÷ √b = √(a ÷ b), and the same laws hold for cube roots (∛a × ∛b = ∛(ab)). We use them to take a perfect-square factor out of a root: √32 = √(16 × 2) = 4√2, and a perfect-cube factor out of a cube root: ∛24 = ∛(8 × 3) = 2∛3. Take out the LARGEST perfect square: √48 = 4√3, not 2√12 (12 still holds a square factor). Simplify first and then add: √32 − √18 + √50 = 4√2 − 3√2 + 5√2 = 6√2. To remove a root from a denominator multiply top and bottom by that root: 5/√3 = 5√3/3. But never write √(a + b) as √a + √b.
Simplify √50 + √8 − √18. √50 = 5√2, √8 = 2√2 and √18 = 3√2, so the sum is (5 + 2 − 3)√2 = 4√2.
A square has side 3√2 cm. Area: (3√2)² = 9 × 2 = 18 cm². Rationalise 6/√2 = 6√2/2 = 3√2. A cube root quotient: ∛40 ÷ ∛5 = ∛8 = 2.
Zaid says √48 = 2√12. Is that simplest form? 2√12 has the same value as √48, but 12 still has the square factor 4: 2√12 = 2 × 2√3 = 4√3. The simplest form is 4√3.
Class activity
Factor tree. Write 72 as a product of its largest perfect square and another number. Then simplify √72. Do the same with √80, √90 and √98. Check each with a calculator.
