What are surds and how do you rationalize a denominator with a single surd or a binomial surd?

The whiteboard discusses the topic of Surds, specifically focusing on Division or Rationalization of Surds.

Here's an explanation of the concepts and solutions to the examples shown:

1. What is a Surd?
A surd is an irrational number that can be expressed as the root of an integer, where the root cannot be simplified to a whole number. For example, $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ are surds, but $\sqrt{4}$ is not a surd because $\sqrt{4} = 2$.

2. Rationalization of Surds
Rationalization is the process of converting a fraction with an irrational number (a surd) in its denominator into an equivalent fraction with a rational number in its denominator. This is done to simplify expressions and make calculations easier, as it's generally preferred not to have surds in the denominator.

3. How to Rationalize a Denominator with a Single Surd
If the denominator is a single surd, like $\sqrt{a}$, you multiply both the numerator and the denominator by that surd. This is because $\sqrt{a} \times \sqrt{a} = a$, which is a rational number.

4. Conjugate of a Surd
The conjugate of a surd expression is used when the denominator is a binomial (two terms) involving a surd, such as $a + \sqrt{b}$ or $\sqrt{a} + \sqrt{b}$. The conjugate is formed by changing the sign between the terms.

• The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$.
• The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$. When an expression is multiplied by its conjugate, the result is a rational number, because $(x+y)(x-y) = x^2 - y^2$. For example, $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b$.

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Examples from the Whiteboard:

Here's how to rationalize the denominators for the given examples:

a) $\frac{1}{\sqrt{2}}$
To rationalize, multiply the numerator and denominator by $\sqrt{2}$:

$$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}$$

The rationalized form is $\boxed{\frac{\sqrt{2}}{2}}$.

b) $\frac{1}{\sqrt{3}}$
To rationalize, multiply the numerator and denominator by $\sqrt{3}$:

$$\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{3}}{3}$$

The rationalized form is $\boxed{\frac{\sqrt{3}}{3}}$.

c) $\frac{1}{\sqrt{5}}$
To rationalize, multiply the numerator and denominator by $\sqrt{5}$:

$$\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{5}$$

The rationalized form is $\boxed{\frac{\sqrt{5}}{5}}$.

d) $\frac{1}{\sqrt{7}}$
To rationalize, multiply the numerator and denominator by $\sqrt{7}$:

$$\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{1 \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{\sqrt{7}}{7}$$

The rationalized form is $\boxed{\frac{\sqrt{7}}{7}}$.

e) $\frac{1}{\sqrt{11}}$
To rationalize, multiply the numerator and denominator by $\sqrt{11}$:

$$\frac{1}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{1 \times \sqrt{11}}{\sqrt{11} \times \sqrt{11}} = \frac{\sqrt{11}}{11}$$

The rationalized form is $\boxed{\frac{\sqrt{11}}{11}}$.