Numbers and operations · Ages 13–14
Surds & irrational numbers
From “Square, cube and nth roots” to “Constructing lengths √n”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- Square, cube and nth rootsⁿa is the number whose nth power is a; √ takes the positive square root
Method∛−125 = −5, because (−5)3 = −125
Another way: Primes, each index divided by n324 = 22 × 34, 324 = 2 × 32 = 18
- Rational and irrational numbersA ratio of integers is rational; n is rational only when n is a perfect square
Method49 = 7 is rational, 50 is irrational
Another way: Between two perfect squares49 < 50 < 64, so 7 < 50 < 8
- Simplifying surdsSplit off the biggest perfect square and take its root outside
Method72 = 36 × 2 = 62
Another way: Primes out in pairs72 = 2 × 2 × 2 × 3 × 3: a pair of 2s and of 3s out, 62
- Working with surdsOnly like surds add; outside times outside, inside times inside; rationalise a root in the denominator
Method12 + 27 = 23 + 33 = 53
Another way: Write the whole number as two roots63 = 2 × 3 × 3 ÷ 3 = 23
- Constructing lengths √nShorter sides a and b give a longest side a2 + b2
Method22 + 32 = 4 + 9 = 13
Another way: The spiral, adding 1 each time2 → 3 → 4 → 5 → 6 → 7
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1Square, cube and nth roots →
25 = 5
True or false?
Show the answer and working
Answer: True
25 has two square roots, 5 and −5, but the sign 25 means the positive one only, 5.
Example 2Simplifying surds →
In simplest form: 30 × 150 =
Fill in each blank
Show the answer and working
Answer: 30, 5
30 × 150 = 4500, and 4500 = 302 × 5, so 305.
Example 3Constructing lengths √n →
A right-angled triangle's two shorter sides are 7 and 4. Its longest side is .
Show the answer and working
Answer: 65
The longest side squared = 72 + 42 = 65, so it is 65.
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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