The process of Rationalisation often involves converting an expression with an irrational denominator into an equivalent expression with a rational denominator. To do this, we must first understand the difference between rational and irrational numbers under a root:
Rational Numbers (under a root): A number whose root can be simplified to a whole number or a fraction.
Example 1: $\sqrt{4} = \sqrt{2 \times 2} = 2$ (A rational number)
Example 2: $\sqrt{16} = \sqrt{4 \times 4} = 4$ (A rational number)
Example 3: $\sqrt{32} = \sqrt{2 \times 2 \times 2 \times 2 \times 2} = 4\sqrt{2}$ (This simplifies to $4 \times (\text{irrational number})$, so the result is irrational. The notes in the image are slightly confusing here, but $\sqrt{32}$ itself is an irrational number because it is $\approx 5.657$. For it to be rational, it must simplify to an integer or fraction. Let’s assume the context meant $\sqrt{32}$ can be simplified, but the final number $\sqrt{32}$ is $\mathbf{irrational}$.)
Irrational Numbers (under a root): A number whose root cannot be expressed as a simple fraction, resulting in non-terminating, non-repeating decimals.
Example: $\sqrt{3}$ is not a rational number.
Example: $\sqrt{5}$ is not a rational number.
A Surd is a specific type of irrational number.
Definition:
A number of the form $\sqrt[n]{a}$ is called a pure surd if:
$a$ is a positive rational number ($a \in \mathbb{R}^+$).
$\sqrt[n]{a}$ is an irrational number.
$n$ is a natural number ($n \in \mathbb{N}$) and $n > 1$.
Examples of Pure Surds:
Important Note: Every surd is an irrational number, but every irrational number is not a surd. For example, $\pi$ and $e$ are irrational numbers, but they cannot be expressed as $\sqrt[n]{a}$ where $a$ is rational, so they are not surds.
Surds are often combined with rational numbers, giving rise to different classifications:
These are surds that are a combination of a rational number and an irrational surd.
Form: $a + \sqrt{b}$ or $a\sqrt{b}$ (where $a$ is rational and $\sqrt{b}$ is a surd).
Examples:
These are pairs of surds that, when multiplied, result in a rational number. They are essential for the process of rationalisation.
Definition:
Two surds of the form $a+\sqrt{b}$ and $a-\sqrt{b}$ are called conjugate surds to each other.
Example:
When multiplied: $(2+\sqrt{3})(2-\sqrt{3}) = 2^2 – (\sqrt{3})^2 = 4 – 3 = 1$ (A rational number).
To work with surds, we use a set of rules based on the laws of exponents. (Assume $a > 0$ and $b > 0$).
Example: $(\sqrt{2})^2 = 2$
Example: $\sqrt[4]{a} = a^{\frac{1}{4}}$