Surds & irrational numbers · Ages 13–14
Square, cube and nth roots
ⁿa is the number whose nth power is a; √ takes the positive square root.
How to do it
Method
ⁿ√a is the number whose nth power is a; √ takes the positive square root
∛−125 = −5, because (−5)3 = −125
Another way
Primes, each index divided by n
324 = 22 × 34, 324 = 2 × 32 = 18
In full
A square root undoes a square: 72 = 49, so the square roots of 49 are 7 and −7, since (−7)2 is 49 too. The sign 49 means the positive one only: 49 = 7. Cube roots: 3125 = 5, because 53 = 125, and negative numbers have cube roots too: ∛−125 = −5, because (−5)3 = −125. nth roots are the same: 481 = 3, because 34 = 81, and ⁵√−32 = −2. A negative number has no square root, nor any even root, among the real numbers, because no number to an even power is negative. For a big number, break it into primes and divide each index by n: 324 = 22 × 34, so 324 = 2 × 32 = 18. Or estimate: 182 = 324; a last digit 4 comes from squaring 2 or 8, then check the size.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
529 =
Show the answer and working
Answer: 23
232 = 529, so 529 = 23 (√ means the positive square root).
Example 2
576 = 2□ × 3□, so 576 = 2□ × 3□ =
Fill in each blank
Show the answer and working
Answer: 6, 2, 3, 1, 24
In primes, 576 = 26 × 32; for the square root halve each index: 23 × 31 = 24.
Example 3
3343 = 7
True or false?
Show the answer and working
Answer: True
(7)3 = 343: a cube root has the number's own sign, so 3343 = 7.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.400 = −20
True or false?
Answer:
2.3125 =
Answer:
3.900 =
Answer:
4.41296 =
Answer:
5.1296 = 2□ × 3□, so 1296 = 2□ × 3□ =
Fill in each blank
Answer:
All the answers
- 1. False
- 2. 5
- 3. 30
- 4. 6
- 5. 4, 4, 2, 2, 36
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