Surds are irrational numbers that cannot be expressed as exact decimals or fractions. They are usually represented as square roots of non-perfect square numbers.
When working with surds, it’s important to simplify expressions by factoring out perfect squares and rationalising denominators when necessary. By doing this, surds can be presented in a simpler and more manageable form, making them easier to work with in mathematical problems.
Let’s look at how to simplify and rationalise surds.
To simplify a surd, we use the rule: .
Consider the following examples:
Now, let’s simplify an expression involving surds:
Ok, let’s look at some more examples.
When working with surds (square roots of non-perfect squares), sometimes we need to rationalise the denominator, which means removing the surds from the denominator. Let’s look at how to rationalise fractions containing surds in their denominators.
The conjugate of is , and the conjugate of is . Conjugates are a useful tool for rationalising surds. To illustrate the process, let’s look at some examples.
Keep in mind this key formula when working with conjugates:
It’s also important to remember this formula:
Rationalise the following: