Number Theory. Argue Is not the square of an integer.

[celtics777]

Homework Statement

Argue that (17^4)*(5^10)*(3^5) is not the square of an integer.

Homework Equations

N/A?

The Attempt at a Solution

Do I break these up, and show that each is not a square? I'm not sure if that would be correct, but sqrt(17^4)=289 * sqrt(5^10)=3125 * sqrt(243)=15.5884 =...

Since sqrt(243)=15.5884 and is not an integer then the above is not the square of an integer. Is this an efficient explanation?

 

[HallsofIvy]

Well, it certainly doesn't show any understanding of the problem! Look at the exponents: \sqrt{17^4}= (17^4)^{1/2}= 17^2. \sqrt{5^{10}}= (5^{10})^{1/2}= 5^5. What about \sqrt{3^5}?

Do you see why the fact that 3, 5, and 17 are prime numbers is important?
(Consider the same question about \sqrt{(8)(18)}.)