JENECONKMaths Universe

Simplifying Surds

Ages 14–18

Difficulty adapts to you

Simplifying Surds

To simplify √n, look for the largest perfect-square factor of n. Use √(ab) = √a × √b, then replace the square root of the perfect square with its whole-number root. Repeat until no square factor greater than 1 remains inside the root. A simplified surd has no square factor left inside its radical.

Your goals

  • ✓You can take square factors out of a surd and leave the simplest exact form.
  • ✓You can use radicand, square factor, simplest form correctly.
  • ✓You can find the largest perfect-square factor, take its root outside, and check by squaring back.

Words to know

radicand:
The number inside a root sign.
square factor:
A factor that is a perfect square, such as 4, 9, or 25.
simplest form:
A form with no square factor greater than 1 left inside the root.

Examples

  • 1√12 = √(4×3) = √4 × √3 = 2√3.
  • 2√18 = √(9×2) = 3√2.
  • 3√72 = √(36×2) = 6√2. Using the largest square factor gives the shortest answer.
  • 4√50 = √(25×2) = 5√2, not 25√2.
  • 5To check 2√3, square it: (2√3)² = 4×3 = 12, so it matches √12.

More than one way to solve it

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

Method 1

Largest square factor

Best for: Simplifying a root efficiently
  1. 1.Find the largest perfect-square factor.
  2. 2.Split the root into two roots.
  3. 3.Take the square root of the perfect square.

Example: √72 = √(36×2) = 6√2.

Method 2

Square-back check

Best for: Checking a simplified surd
  1. 1.Square the outside number.
  2. 2.Square the root part.
  3. 3.Confirm the product is the original radicand.

Example: (2√3)² = 4×3 = 12, so 2√3 matches √12.

Check these

  • • Do not take the square root of each factor incorrectly or leave a square factor inside the 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 simplifying surds example, solve it, and explain how you checked it.

Byte

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Helpful Hints

“If you are stuck, skip and come back later.”