Define rational and irrational numbers and perform operations with surds and fractional indices (ACMNA264)
LO: To define rational and irrational numbers and perform operations with surds and fractional indices.
Know:
 What is a rational number
 What is an irrational number
 How to complete the four basic operations
 What is a surd
 What is an indices
Understand:
 That certain numbers can’t be simplified to remove a square root and that operations can be used to solve problems involving surds.
Do:
 I can define rational and irrational numbers and perform operations with surds and fractional indices.
Examples (Visual Representations)
Notes
What are surds?
Surds are square roots which can’t be reduced to rational numbers (exact numbers).
For example root25 = 5. This is NOT a surd because the result is a whole number (rational number).
root5 = 2.2360679775. This is a surd because the result is not a rational number (the number goes on forever and doesn’t have an exact value).
Simplifying Surds
Simplifying surds are pretty easy.
The first step is to break the original number into it’s factors. (HINT: It’s a little bit easier if you find square numbers for factors)
Then determine which factors are surds and which are not surds.
Nonsurd numbers can then be removed and put in front of the square sign.
In the example to the left, we have the square root of 50. 50 can be broken down to 25 x 2. We know that the square root of 25 is 5 so we can move the 5 in front of the square root sign and we are left with the square root of 2 as a surd.
Multiplying Surds
Multiplying surds are also pretty easy.
The first step is to multiply any numbers in front of the square root sign.
Then multiply the numbers under the square root sign.
Then simplify the number as much as you can.
In the example to the left, we have 5root2 multiplied by 3root22.

multiply 5 by 3 which gives us 15.

multiply root 2 by root 22 which gives us root 44.

root 44 can be simplified into root4 multiplied by root11

root 4 can be simplified into 2

2 can then be multiplied to our original 15, which gives us 30.

which leaves us with an answer of 30root11.