Surds & irrational numbers · Ages 13–14
Simplifying surds
Split off the biggest perfect square and take its root outside.
How to do it
Method
Split off the biggest perfect square and take its root outside
72 = 36 × 2 = 62
Another way
Primes out in pairs
72 = 2 × 2 × 2 × 3 × 3: a pair of 2s and of 3s out, 62
In full
Multiplying surds: a × b = ab; dividing: a ÷ b = a ÷ b, for a and b positive. To simplify n, split n into the biggest perfect square that divides it times what is left, and take the square's root out: 72 = 36 × 2 = 36 × 2 = 62. In simplest form no square factor is left under the root: 218 isn't finished, since 18 = 9 × 2. Another way: break n into primes and bring one of each pair out. 72 = 2 × 2 × 2 × 3 × 3: a pair of 2s and a pair of 3s give 2 × 3 = 6 outside, and the single 2 stays inside, 62. The other way round, 62 = 36 × 2 = 72, which helps to compare sizes.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In simplest form: 50 =
Fill in each blank
Show the answer and working
Answer: 5, 2
50 = 25 × 2, 25 the biggest square factor: 50 = 25 × 2 = 52.
Example 2
In simplest form: 296 =
Fill in each blank
Show the answer and working
Answer: 8, 6
96 = 46, so 296 = 2 × 46 = 86.
Example 3
27 = 28
True or false?
Show the answer and working
Answer: True
To bring 2 back under the root, square it: 27 = 22 × 7.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In simplest form: 384 =
Fill in each blank
Answer:
2.63 = 47
True or false?
Answer:
3.18 × 32 =
Answer:
4.In simplest form: 24 × 100 =
Fill in each blank
Answer:
5.43 =
Answer:
All the answers
- 1. 8, 6
- 2. False
- 3. 24
- 4. 20, 6
- 5. 48
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake