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To Identify Perfect Roots, Multiply, Guess, Memorize, or Factorize

Date: 06/21/2011 at 11:09:00
From: Lucretia 
Subject: a question of determining cube roots

Is there a way to figure out what the cube root of a particular number is?

For example, I had this problem:

   x^3 + 125
   
I knew it was a sum of two cubes, but I only determined that 125 = 5^3
after getting lost for a long time trying

       125/3 = ?
       5/125 = ?
       5 * 5 = 25
   5 * 5 * 5 = 125

If it is a larger number, how do you know where to even start? For 512, I
tried 2, because it is even. It took me 45 minutes to figure out it was 8.

Also ... if you have time ... could you give me a hint on how to figure
the same thing for squares? If you have any number, how do you check if it
is a square of some other number?

I am just unfamiliar with the whole idea. I know that dividing a number by
itself will be 1, but division doesn't seem to work. How do you figure out
which number you multiply by itself 3 times to get numbers like 125 and
512?

I am an older citizen who has returned to school recently. I really want
my diploma. I have to pass an algebra class and I have no idea what I am
doing. I didn't have algebra in middle school and didn't do well in math
at all in elementary school, so I am pretty intimidated. Any help at all
would be greatly appreciated!

Thank you. 



Date: 06/21/2011 at 14:40:43
From: Doctor Peterson
Subject: Re: a question of determining cube roots

Hi, Lucretia.

There are two main ways today to find a cube or square root. One is to use
a scientific calculator; you should probably have one, and in fact any
computer should have a calculator application that will do what you need.
Let me know what kind you have, and I should be able to give you
directions for using it.

But it's a good idea to also be able to recognize small cubes and squares
on sight, which is what you are expected to do in problems like this. I
recommend making a table of perfect squares and cubes, which you can refer
to as needed.

The list below contains the values I'd expect a student to know:

    n   n^2   n^3
   ---  ---   ---
     2    4     8
     3    9    27
     4   16    64
     5   25   125
     6   36
     7   49
     8   64
     9   81
    10  100  1000
    11  121
    12  144

Try to familiarize yourself with these values, so that you don't have to
keep refering to a memory aid.

You should also know that the square of a negative number is positive, so
that the squares shown (n^2) are the same values you'd get by squaring -2,
-3, and so on; and that the cube of a negative number is the negative of
the cube of its absolute value. All of that means you don't need to
separately remember the following table:

    n   n^2   n^3
   ---  ---   ---
    -2    4    -8
    -3    9   -27
    -4   16   -64
    -5   25  -125
    -6   36
    -7   49
    -8   64
    -9   81
   -10  100 -1000
   -11  121
   -12  144

Knowing these facts, you can immediately recognize that

   x^3 + 125

... is ...

   (x)^3 + (5)^3

From this insight, you can then use the formula for the sum of cubes.

There are two other methods you might use when the number is not on this
list, such as your example of 512.

First, you can use "guess and check": You know that the cube of 5 is 125
and the cube of 10 is 1000, so the cube root of 512 has to be between 5
and 10. Just try whole numbers in this small range, perhaps starting at
the top (because cubes increase fast):

   9^3 = 9*9*9 = 729
   8^3 = 8*8*8 = 512

Second, you can use the prime factorization of a number. If you factor
512, you find that it is 

   2*2*2*2*2*2*2*2*2 = 2^9.

The exponent is a multiple of 3, so 512 is a perfect cube; and in
particular the cube root is 2^3, the exponent being 1/3 of the exponent
given. If you've learned about simplifying radicals, that area of
knowledge can help here.

But if these last two methods fail, or are too much to do, just get out
your calculator!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Square Roots

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