Surds

Surds

A surd is an expression that includes a square root, cube root or other root symbol.

Surds are used to write irrational numbers precisely.

Leaving the square root sign in is useful for two reasons:

- We don´t need calculators with dealing with them

- They are much more accurate as you don´t have to round off the decimal

Multiple Select

For example: A square has an area of 3 $m^2$  . 

Indicate the exact length of the side of the square.

• 1,73...... m

•  $\sqrt{3}\ m$  

This answer is in surd form. It is irrational and it is said to be "in exact form". A decimal answer, such as 1.73 (2 decimal places), is not exact.

Even 1.732050807568877 is not exact. When an answer is required in exact form, you must write it as a surd, ideally simplifying it if possible.

Multiple Select

Which of the following are surds?

• $\frac{\pi}{2}$

• $\sqrt{5}$

• $\left(\sqrt{17}\right)^2$

• $\sqrt{3}$

• $\sqrt{2}+1$

The following rules apply:

•  $\sqrt{a}.\sqrt{b}=\sqrt{a.b}$  

•  $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$  

Examples:

•  $\sqrt{2}.\sqrt{3}=\sqrt{6}$  

•  $\frac{2\sqrt{27}}{\sqrt{3}}=2\sqrt{\frac{27}{3}}=2\sqrt{9}=2.3=6$  

Simplifying surds:

Surds can be simplified if the number in the root symbol has a square number as a factor.

Example: $\sqrt{12}$               Notice that 12 = 4 . 3 

 $\sqrt{12}=\sqrt{4\ .\ 3}=\sqrt{4}.\sqrt{3}=2\sqrt{3}$  

Another example:

 $3\sqrt{50}$  

50 = 25 . 2  

 $3\sqrt{50}=\sqrt{5\ \ .\ \ 5\ .\ \ 2}=3\sqrt{5^2}.\ \sqrt{2}=3\ .\ 5\ .\ \sqrt{2}=15\sqrt{2}$  

Simplify:

Use rules to simplify surds: