Multiplying surds uses the product law √a × √b = √(ab); dividing uses the quotient law √a ÷ √b = √(a/b). Both rules let you combine or split surds without reaching for a calculator, keeping every answer in exact form — a key skill across GCSE algebra, geometry, and trigonometry.
What are the product and quotient laws for surds?
The two fundamental rules are:
- Product law: √a × √b = √(ab), provided a ≥ 0 and b ≥ 0.
- Quotient law: √a ÷ √b = √(a/b), provided a ≥ 0 and b > 0.
These follow directly from index law: √a = a^(1/2), so a^(1/2) × b^(1/2) = (ab)^(1/2) = √(ab).
A special case worth remembering: √a × √a = a. Squaring removes the surd entirely.
How do you multiply surds together?
Step 1 — Multiply the numbers outside the square root signs together.
Step 2 — Multiply the numbers inside the square root signs together using the product law.
Step 3 — Simplify the result if possible (look for square factors).
Worked example 1: calculate √3 × √12
√3 × √12 = √(3 × 12) = √36 = 6
No surd remains because 36 is a perfect square.
Worked example 2: calculate 2√5 × 3√7
Multiply the integers: 2 × 3 = 6.
Multiply the surds: √5 × √7 = √35 (no square factors, so it stays).
Answer: 6√35
Worked example 3: calculate √8 × √6
√8 × √6 = √48 = √(16 × 3) = 4√3.
Answer: 4√3
Always check whether the product has a perfect square factor and simplify.
How do you divide surds?
Step 1 — Divide the integers.
Step 2 — Divide the surds using the quotient law.
Step 3 — Simplify.
Worked example 4: calculate 6√20 ÷ 3√5
Divide integers: 6 ÷ 3 = 2.
Divide surds: √20 ÷ √5 = √(20/5) = √4 = 2.
Answer: 2 × 2 = 4
The result is an integer — always worth checking.
Worked example 5: calculate √50 ÷ √2
√50 ÷ √2 = √(50/2) = √25 = 5
How do you expand brackets containing surds?
Use exactly the same method as expanding algebraic brackets — multiply every term inside by the term outside.
Worked example 6: expand √2(3 + √8)
√2 × 3 = 3√2.
√2 × √8 = √16 = 4.
Answer: 3√2 + 4
Worked example 7: expand and simplify (1 + √3)(2 + √3)
Use FOIL (or the grid method):
First: 1 × 2 = 2.
Outer: 1 × √3 = √3.
Inner: √3 × 2 = 2√3.
Last: √3 × √3 = 3.
Collect: 2 + √3 + 2√3 + 3 = 5 + 3√3
Worked example 8: expand (3 + √5)(3 − √5)
This is a difference of two squares: (a + b)(a − b) = a² − b².
(3)² − (√5)² = 9 − 5 = 4
Notice: multiplying conjugate surd pairs always eliminates the surd — this is the mechanism behind rationalising the denominator.
Summary table of surd multiplication and division rules
| Operation | Rule | Example | Result |
|---|---|---|---|
| √a × √b | = √(ab) | √3 × √27 | √81 = 9 |
| √a × √a | = a | √7 × √7 | 7 |
| k√a × m√b | = km√(ab) | 2√3 × 5√3 | 30 |
| √a ÷ √b | = √(a/b) | √72 ÷ √2 | 6 |
| (a + √b)(a − √b) | = a² − b | (4 + √3)(4 − √3) | 13 |
How does this connect to rationalising the denominator?
When you divide by a surd — for example, 1/√5 — you multiply numerator and denominator by √5:
1/√5 × √5/√5 = √5/5.
This uses the quotient law and the identity √5 × √5 = 5. Multiplying by a conjugate pair (e.g. (2 + √3)) uses the difference-of-two-squares expansion above. These connections mean the three surd skills — simplifying, multiplying/dividing, and rationalising — form one unified toolkit.
Frequently asked questions
Can I only multiply surds that have the same number under the root?
No. Unlike adding and subtracting surds (which require the same radicand), you can multiply or divide any two surds using the product and quotient laws: √5 × √7 = √35. The only restriction is that the values under the root must be non-negative.
What if the product gives a very large number under the root?
Always look for perfect square factors before declaring the answer. If you get √(200), look for the largest perfect square factor: 200 = 100 × 2, so √200 = 10√2. Practise prime factor decomposition (breaking the number into prime factors) to spot these efficiently.
Why does √a × √a equal a and not √(a²)?
Both are the same: √(a²) = a (for a ≥ 0), because squaring and square-rooting are inverse operations. This also connects to index law: a^(1/2) × a^(1/2) = a^(1) = a. Either way of thinking gives the same result.
Will these rules appear in both non-calculator and calculator papers?
Yes. Simplifying, multiplying, and dividing surds are tested primarily in the non-calculator paper, where the expectation is that answers are left in exact surd form. The calculator paper may use surds within a longer question — for example, as lengths in a Pythagoras or trigonometry problem — and ask you to show exact working.
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