What's the cube root of a number?


The cube root of a number n (³√n) is that figure which cubed (i.e. multiplied by itself twice) gives n.


Remember ³√ is the cube root symbol.


If n is a cube number (perfect cube) (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 etc), its cube root is a whole number.


But if n is a non-cube number (2, 3, 4, 5, 6, 7, 9, 10, 11, 12 etc), its cube root is a decimal and less easy to calculate.


Let's consider the cube roots of cube numbers first.


How to work out the cube root of 1


Which figure cubed (times by itself twice) gives 1?


The answer's 1 because

1³   =   1  x  1  x  1   =   1


Therefore



Cube root of 8


What figure cubed (multiplied by itself twice) gives 8?


The answer is 2

2³   =   2  x  2  x  2   =   8


So



Cube root of 27


Which figure cubed (times by itself twice) gives 27?


The answer is 3

3³   =   3  x  3  x  3   =   27


So



Now what are the cube roots of the remaining cube numbers up to 1000?


The workings out are as follows


  • ³64 = 4  (4³ = 4 x 4 x 4 = 64)
  • ³125 = 5  (5³ = 5 x 5 x 5 = 125)
  • ³216 = 6  (6³ = 6 x 6 x 6 = 216)
  • ³343 = 7  (7³ = 7 x 7 x 7 = 343)
  • ³512 = 8  (8³ = 8 x 8 x 8 = 512)
  • ³729 = 9  (9³ = 9 x 9 x 9 = 729)
  • ³1000 = 10  (10³ = 10 x 10 x 10 = 1000)


Notice all these cube roots are whole numbers.


Next take a look at the cube roots of non-cube numbers


On your calculator, enter 2 followed by the cube root (³) button



- you may need to press a shift or 2nd function key to access ³


The result is

1.25992105


You've calculated ³√2 - the cube root of 2.


It's a decimal, as are the cube roots of other non-cube numbers. For example, use your calculator to check the following


  • ³√3   =   1.44224957
  • ³√4   =   1.587401052
  • ³√5   =   1.709975947
  • ³√6   =   1.817120593
  • ³√7   =   1.912931183
  • ³√9   =   2.080083823
  • ³√10   =   2.15443469
  • ³√11   =   2.223980091
  • ³√12   =   2.289428485


Feeling more confident about cube roots?


After reading carefully through the examples provided in this lesson, you should now be able to work out cube roots regardless of whether they're whole numbers or decimals - albeit sometimes using your calculator!


For related topics click:

Cube numbers

Square numbers

Square roots


Back to top


› Cube roots