JENECONKMaths Universe

Adding and Subtracting Surds

Ages 14–18

Difficulty adapts to you

Adding and Subtracting Surds

You can combine surds only when their root parts match, just as you combine like algebraic terms. Simplify every surd first. Then add or subtract the numbers outside the matching root. Unlike surds, such as √2 and √3, cannot be combined into one surd.

Your goals

  • ✓You can combine like surds after simplifying each term.
  • ✓You can use like surds, unlike surds, coefficient correctly.
  • ✓You can simplify first, then combine only terms with the same root part.

Words to know

like surds:
Surd terms with the same simplified root part.
unlike surds:
Surd terms with different root parts that cannot be combined.
coefficient:
The number multiplying the surd, such as 3 in 3√2.

Examples

  • 12√3 + 5√3 = 7√3 because both terms contain √3.
  • 2√12 + √3 = 2√3 + √3 = 3√3. Simplify before deciding whether terms match.
  • 33√5 − √20 = 3√5 − 2√5 = √5.
  • 4√2 + √3 stays as √2 + √3; these are unlike surds.
  • 5Check by approximation: 2√3 + √3 is about 3(1.732) = 5.196, consistent with 3√3.

More than one way to solve it

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

Method 1

Collect like roots

Best for: Adding and subtracting surds
  1. 1.Simplify every root first.
  2. 2.Group terms with the same root part.
  3. 3.Add or subtract only the outside numbers.

Example: √12 + √3 = 2√3 + √3 = 3√3.

Method 2

Unlike-surds decision

Best for: Knowing when to stop
  1. 1.Compare the simplified root parts.
  2. 2.If they differ, do not combine them.
  3. 3.Leave the exact sum as it is.

Example: √2 + √3 cannot be reduced to one surd.

Check these

  • • Do not add the numbers inside different roots or combine unlike surds.
  • • 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 adding and subtracting surds example, solve it, and explain how you checked it.

Professor Wise Owl

Professor Wise Owl

Wisdom & Understanding

“Every expert was once a beginner. You are on your way!”