## Surds

Surds are special types of irrational number, such as and . Some surds give rational values, such as , however others do not have an exact decimal value.

*Think:* What is the number between and . Is that number ? Why not?

**Basic Properties of Surds**

### Examples:

Addition and Subtraction

Treat surds in the same way as pronumerals in algebra. Only add or subtract 'like surds' the same way 'like terms' are calculated.

1. Simplify

2. Simplify

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Multiplication and Division

3. Simplify

4. Simplify

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5. Simplify

6. Simplify

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## Expanding Brackets

The same rules for expanding brackets and binomial products that you use in algebra also applies to surds. Go to Group Symbols and Binomials to find out more.

**Properties of Surds as Binomials**

### Examples:

Expand and simplify:

1.

2.

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(This example is called the **difference of two squares**.)

## Rationalising the Denominator

Rationalising the denominator of a fractional surd means writing it with a rational number (not a surd) in the denominator. For example, after rationalising the denominator, becomes .

Back when calculators were available, rationalising denominator is used to make it easier to divide a fraction by a rational number than an irrational one.

When there is a binomial denominator, we use the difference of two squares to rationalise it, as the result is always a rational number.

The sum and difference of two simple quadratic surds are said to be **conjugate surds** to each other.

Conjugate surds are also known as **complementary surds**.

Two surds and are **conjugate** to each other.

To rationalise the denominator of , multiply by .

### Examples:

1. Rationalise the denominator of

2. Rationalise the denominator of

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3. Write with a rational denominator:

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4. Evaluate as a cfraction with rational denominator:

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