Surds

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.

Non-surd 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.

  1. multiply 5 by 3 which gives us 15.

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

  3. root 44 can be simplified into root4 multiplied by root11

  4. root 4 can be simplified into 2

  5. 2 can then be multiplied to our original 15, which gives us 30.

  6. which leaves us with an answer of 30root11.

Adding and Subtracting Surds

Adding surds are also pretty easy.

The first step is to identify like surds. Like surds are numbers which have the same number under the square root sign. This is very similar to collecting like terms when working with algebraic expressions.

In the first scenario, since both numbers have a square root of 2, all we have to do is add the numbers in front of the square root sign. 5root2 + 7root2 = 12root2.

Same can be completed with subtraction.

Student Exemplars

SURDS Videos

Surds Resources

BBC Bitesize Surds
Maths is Fun – Surds
Next Lesson – Logs

Practice Questions

My Maths 10/10A

pg. 105 Exercise 3A-1 Q. 1,2

pg. 106 Exercise 3A-2 Q. 1-9

pg. 111 Exercise 3B Q. 1-9

pg. 117 Exercise 3C Q. 1-8

pg. 123 Exercise 3D Q. 1-9

pg. 130 Exercise 3E Q. 1-7

pg. 135 Exercise 3F Q. 1-4