Is the square root of 945 irrational?

[srfriggen]

Is the square root of 945 irrational?

I feel it is rational because my TI-84 Plus converts it into 275561/8964, however, I am unsure whether the calculator is estimating.

Can someone please advise. It can be broken down into 3√105, and again, my calculator is able to convert √105 into a fraction.

Thank you.

 

[phyzguy]

Yes, sqrt(945) is irrational. Your calculator is finding a fraction approximating sqrt(945).

sqrt(945) = 30.740852297878796...
275561/8964 = 30.740852298081215...

They are not equal.

 

[mathman]

If the square root of an integer is rational it must be an integer. The square of this fraction is 945.000000012445056. You are running into the limits on the calculator significant digits.

 

[StoneTemplePython]

I typed these 3 commands into worlfram
https://www.wolframalpha.com/input/?i=prime+factorization+of+8964
https://www.wolframalpha.com/input/?i=prime+factorization+of+275561
https://www.wolframalpha.com/input/?i=prime+factorization+of+945

so supposing it is rational we get

$3^3 \cdot 5 \cdot 7 = 945 = \big(\frac{275561}{8964}\big)^2= \big( \frac{11\cdot 13\cdot 41 \cdot 47}{2^2 \cdot 3^3 \cdot 83 }\big)^2$

clearing the denominator gives

$\big(3^3 \cdot 5 \cdot 7\big)\big(2^2 \cdot 3^3 \cdot 83 \big)^2 = 945\big(2^2 \cdot 3^3 \cdot 83 \big)^2 = \big( 11\cdot 13\cdot 41 \cdot 47\big)^2$

but this violates the fundamental theorem of arithmetic and hence what your calculator gave cannot be an exact rational expression for the square root

 

[mfb]

Two-line approach: 275561/8964 is a fraction that doesn't share prime factors in numerator and denominator. Therefore (275561/8964)² is also a fraction with this property and the numerator cannot be a multiple of the denominator.