Numbers that Are Both Perfect Squares and Perfect CubesDate: 04/10/99 at 12:42:02 From: Ben Houge Subject: perfect squares, perfect cubes Hi -- I am supposed to find two integers that are both a perfect square and a perfect cube. I know what a perfect number is, but I'm not sure about a perfect square or perfect cube. I have looked in my 8th grade textbook, which defines perfect number but not perfect square or cube. I've looked in a regular dictionary, a CD encyclopedia, and I've looked through Dr. Math online, too. Could you define those terms for me? Thank you! Date: 04/11/99 at 00:02:57 From: Doctor Bruce Subject: Re: perfect squares, perfect cubes I sympathize with your confusion -- the word "perfect" is being used in two very different ways! A perfect number is a whole number the divisors of which (including 1 but not the number itself) add up exactly to the number. That should be the definition you found in your textbook. But perfect squares and perfect cubes are different. A perfect square means simply a whole number which is the square of another whole number. Likewise, a perfect cube is a whole number which is the cube of another whole number. For example, 25 is a perfect square (of 5), and 27 is a perfect cube (of 3). It is true that 2 is the square of 1.4142... (a certain decimal number), but we could not say that 2 is a *perfect* square, because the number 1.4142... is not a whole number. You said you wanted to find a number which is a perfect square *and* a perfect cube at the same time. These are called "perfect sixth powers." Some examples of these are: 0, 1, 64, 729, 15625, ... Hope this helps, - Doctor Bruce, The Math Forum http://mathforum.org/dr.math/ |
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