Addition and subtraction of surds

Addition and Subtraction of Surds

Introduction

  • Surds are numbers left in ‘square root form’ (or ‘cube root form’ etc). They are therefore irrational numbers.
  • The value is exact because it is not rounded, unlike a decimal.

Addition and Subtraction Rules for Surds

  • Addition or subtraction of surds is possible only when the surds are like surds or similar surds, meaning the numbers under the square root symbol are the same.
  • Like terms can be added or subtracted just like algebraic expressions, by tallying the coefficients (numbers outside the square roots).

Step-by-Step Guide for Addition and Subtraction of Surds

  1. Identify whether the surds you are dealing with are like surds
  2. If they are, you can add or subtract them by altering the coefficients while the radicand (the number inside the root) remains unchanged.
  3. If surds are different, you can simplify them, or rationalise them, into like surds. Once done, addition or subtraction can take place.

Examples

  • (3\sqrt2 + 2\sqrt2 = 5\sqrt2) because the radicand is the same, meaning the surds are like. Hence, the coefficients 3 and 2 are added.
  • (7\sqrt5 - 3\sqrt5 = 4\sqrt5) by the same principle, as the radicands are same (both are 5), the coefficients 7 and 3 can be subtracted.
  • However, (8\sqrt6 + 2\sqrt3) can’t be simplified as these surds are unlike surds (6 and 3 are different).

Note

  • An important skill when dealing with surds is being able to simplify them and rationalise denominators to make calculation easier. This often involves conversion into like surds to enable addition or subtraction to occur.
  • The same principles of addition and subtraction of surds can be applied to cube roots, fourth roots, etc.