Manipulating Surds

Manipulating Surds

Understanding Surds

  • Surds are numbers left in ‘square root form’ (or ‘3rd root’, ‘4th root’ etc). They are therefore irrational numbers which can’t be represented as a fraction and their decimal representation goes on forever without repeating.
  • Some square roots can simplify into whole numbers; for example, √16 = 4. However, numbers like √2 do not simplify into a whole number and are left as surds.
  • Simple surds can be manipulated into compound surds and vice versa.

Simplifying Surds

  • The simplest form of a surd has no square number (other than 1) which divides exactly into the number under the square root sign.
  • Start by finding the largest square number that divides into the number under the square root sign. Then, apply the laws of surds to simplify it.
  • For example, the surd √18 can be simplified as follows: √18 = √(9x2) = √9 x √2 = 3√2.

Rationalising the Denominator

  • It is generally considered best practice not to have a surd in the denominator of a fraction.
  • Rationalising the denominator is the process of eliminating surds from the denominator. To rationalise a denominator, multiply the top and bottom of the fraction by the surd in the denominator.
  • For example, to rationalise the denominator of 1/√3, multiply top and bottom by √3: 1/√3 = √3/3.

Operations with Surds

  • You can add and subtract surds if they have the same value under the square root sign. For example, 4√3 + 3√3 = 7√3.
  • Surds can be multiplied together using the rule √a x √b = √(ab). For example, √3 x √2 = √6.
  • Similarly, surds can be divided using the rule √a / √b = √(a/b). For example, √50 / √2 = √25 = 5.

Practice Problems

  • Simplify the following surd: √72
    • Solution: √72 = √(36x2) = √36 x √2 = 6√2.
  • Rationalise the denominator of the following fraction: 5/√7.
    • Solution: 5/√7 = 5√7/7.
  • Multiply the following surds: √5 x √20
    • Solution: √5 x √20 = √(5x20) = √100 = 10.

Final Notes

  • Surds are a powerful mathematical tool when dealing with irrational numbers. Make sure to practice different operations with surds, as they are often combined in GCSE exam questions.
  • Understanding how to rationalise the denominator is particularly important for algebraic fractions, where the process can be a little more complex.
  • Practice is key when it comes to manipulating surds. Use a variety of problems to make sure you understand each concept.