Squaring x + y + z = 0 gives x2 + y2 + z2 + 2(xy + yz + zx) = 0. Since xy + yz + zx = 0, it simplifies to x2 + y2 + z2 = 0. ...
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The denominator is 2 cubed times 5, which equals 40. Since the highest power between the prime factors 2 and 5 is 3, the terminating decimal expansion will have exactly 3 decimal places.
First, find a common denominator for 1/6 and 2/5, which is 30, giving 5/30 and 12/30. The five rational numbers lying between them are 6/30, 7/30, 8/30, 9/30, and 10/30.
To find 6 rational numbers between 3 and 4, convert them into fractions with a denominator of 7, giving 21/7 and 28/7. The intermediate numbers are 22/7, 23/7, 24/7, 25/7, 26/7, and 27/7.
To find 5 rational numbers between 2/5 and 3/5, multiply the numerator and denominator of both fractions by 6. This gives 12/30 and 18/30. The intermediate rational numbers are 13/30, 14/30, 15/30, 16/30, and 17/30.