JENECONKMaths Universe

Multiplying and Dividing Surds

Ages 14–18

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Multiplying and Dividing Surds

For multiplication, multiply the numbers outside the roots and multiply the numbers inside the roots: √a×√b = √(ab). Then simplify. For division, divide outside numbers and use √a/√b = √(a/b) when the denominator is non-zero. Always simplify the final surd and keep the denominator rational when required.

Your goals

  • ✓You can multiply, divide and simplify exact surd expressions.
  • ✓You can use conjugates, difference of squares, radical correctly.
  • ✓You can multiply outside and inside parts separately, then simplify the root.

Words to know

conjugates:
A pair such as a+√b and a−√b with the sign changed.
difference of squares:
The identity (a+b)(a−b) = a²−b².
radical:
A root expression such as √2 or √7.

Examples

  • 1√3 × √12 = √36 = 6.
  • 22√5 × 3√2 = 6√10.
  • 3√48 ÷ √3 = √16 = 4.
  • 44√6 ÷ 2√3 = 2√2 after dividing the outside numbers and simplifying √(6/3).
  • 5(√5 + 1)(√5 − 1) = 5 − 1 = 4 by the difference-of-squares pattern.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Outside and inside

Best for: Multiplying surd terms
  1. 1.Multiply the outside numbers.
  2. 2.Multiply the numbers inside the roots.
  3. 3.Simplify the resulting root.

Example: 2√5×3√2 = 6√10.

Method 2

Conjugate product

Best for: Simplifying expressions with opposite signs
  1. 1.Identify a + √b and a − √b.
  2. 2.Multiply as a difference of squares.
  3. 3.Cancel the middle terms.

Example: (√5+1)(√5−1) = 5−1 = 4.

Check these

  • • Do not multiply only the numbers outside the roots or forget to simplify the final root.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own multiplying and dividing surds example, solve it, and explain how you checked it.

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