All real cube roots of 216

Question

I need help answering math problems and I think this will help

Answer 1

THIS MIGHT BE A PROBLEM requiring De Moivre's Theorem (in Trigonomletry class), because it's asking for ALL cube roots.

 

It's usually considered that there are 3 cube roots of a real number, only one of which is real, the other two being complex (of the form a + ib).

 

If not the above type of question, then here's good advice: To find principal (real) square roots or cube roots of numbers, spend a little time memorizing the various power of 2 and 3 so that (for example) 64 = 2⁶ in your mind.

 

Then cube root of 64 = (2⁶)^1/3 = 2² = 4, etc.

Answer 2

Hi Gage,

 

I like making trees to solve problems involving square roots, cube roots, etc.

 

                                 216

                                  /\

                               2 • 108

                                      /\

                                   2 • 54 

                                         /\
                                       2 • 27

                                             /\
                                          3 • 9

                                               /\
                                             3 • 3

 

Notice that each branch ends in a prime number.

Since we need a cube root, we want to find the same prime number appearing 3 times in the tree.  Notice that 2 (bolded) appears 3 times and 3 appears 3 times (italics).  There are no other factors at the ends of the tree.

 

So the cube root of 216 is

2 • 3 = 6

 

You can check this

 

6 • 6 • 6 = 216