Surds & irrational numbers · Ages 13–14
Working with surds
Only like surds add; outside times outside, inside times inside; rationalise a root in the denominator.
How to do it
Method
Only like surds add; outside times outside, inside times inside; rationalise a root in the denominator
12 + 27 = 23 + 33 = 53
Another way
Write the whole number as two roots
63 = 2 × 3 × 3 ÷ 3 = 23
In full
Only like surds, with the same number under the root, add and subtract, as like terms do: 32 + 52 = 82, just as 3x + 5x = 8x. When the numbers under the roots differ, simplify each first and see whether they become alike: 12 + 27 = 23 + 33 = 53. 2 + 3 doesn't combine, and it isn't 5. To multiply, multiply the numbers outside and the numbers inside: 23 × 46 = 818 = 8 × 32 = 242. With a surd in the denominator, multiply the top and the bottom by that surd so the bottom becomes whole: rationalising the denominator. 63 = 6 × 33 × 3 = 63 ÷ 3 = 23. Or write 6 as 2 × 3 = 2 × 3 × 3 first and cancel one 3: the same answer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
410 + 210 =
Fill in each blank
Show the answer and working
Answer: 6, 10
Like surds combine like 4x + 2x: (4 + 2)10 = 610.
Example 2
20 + 80 = ?
- A. 65
- B. 100
- C. 610
- D. 85
Choose the right answer
Show the answer and working
Answer: A. 65
20 = 25 and 80 = 45; added, 65.
Example 3
In simplest form: 58 − 332 =
Fill in each blank
Show the answer and working
Answer: −2, 2
8 = 22 and 32 = 42; 5 × 2 − 3 × 4 = −2, so −22.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.65 + 25 = 810
True or false?
Answer:
2.112 − 52 =
Fill in each blank
Answer:
3.In simplest form: 28 + 252 =
Fill in each blank
Answer:
4.(24 + 4)(24 − 4) =
Answer:
5.In simplest form: 28 − 532 =
Fill in each blank
Answer:
All the answers
- 1. False
- 2. 6, 2
- 3. 8, 7
- 4. 8
- 5. −16, 2
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
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