Simplify numerical expressions involving surds
Surds Basics
- Surds are numbers left in 'square root form' (or 'cube root form' etc). They are often used when the square root of a number doesn't have an exact value.
- A surd can be manipulated with the same mathematical rules as other numerical expressions.
Simplifying Surds
- To simplify a surd, find the largest square number that divides into the number under the square root sign.
- For example, to simplify β18, which can be expressed as β(9*2), it becomes 3β2.
- The process of simplifying surds is called 'rationalising the denominator'.
Multiplying and Dividing Surds
- When multiplying surds, multiply the numbers under the root. For βexample, β2 * β3 = β6.
- When dividing surds, divide the numbers under the root. For example, β8 / β2 = β4 = 2.
Adding and Subtracting Surds
- Surds can be added or subtracted when they are the same type. For example, β3 + β3 = 2β3.
- If the surds are not the same type, they cannot be added or subtracted directly. For example, β2 + β3 cannot be simplified further.
Rationalising the denominator
- When a surd is in the denominator of a fraction, the fraction needs to be manipulated until the surd is removed from the denominator. This process is called 'rationalising the denominator'.
- To rationalise the denominator, multiply the top and bottom of the fraction by the conjugate of the bottom. For example (1/β3) * (β3/β3) = β3/3.