express square root of 4608 in simplest radical form

Question

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Answer 1

Hi John:

One way to do this is to do a prime factor tree of 4608 as below:

 

                                          4608

                                      2  X  2304

                                   2  X  2  X 1152

                                 2  X  2  X  2  X  576

                               2  X  2  X  2  X  2  X  288

                            2  X  2  X   2  X  2  X  2  X  144

                        2  X  2  X  2  X  2  X  2  X  2  X  72

                      2  X  2  X  2  X  2  X  2  X  2  X  2  X  36

                    2  X  2  X  2 X  2  X  2  X  2  X  2  X  2  X  18

                 2  X  2  X  2  X  2  X  2  X  2  X  2  X  2  X  2  X  9

               2  X  2  X   2  X  2  X  2  X  2  X  2  X  2  X  2  X  3  X  3

 

4608 = (2⁹)(3²)

 

√4608 = √[2⁹(3²)] = √[2⁸(2)(3²)] = √[(2⁴)²(2)(3²)] = 2⁴(3)√2 = 48√2

 

The answer is 48√2

 

Check via calculator:  48√2 = 67.88251 ...  √4608 = 67.88251  Answer is correct

 

 

                  

Answer 2

Divide 4608 by a perfect square to find an integer factor.  We do this because we want to find two integers that multiply to 4608, in which one of the factors is a perfect square.

 

√32 * √144

 

 

Now we can break it down further.

 

√144 * √16 * √32 =

 

12 * 4 * √32 =

 

48√32