Puzzles solutions

1 [(2+√2)/(2-√2)] can be written with a rational denominator by multiplying top and bottom by (2 + √2), giving [(2 + √2)(2+√2)]/[(2-√2)(2+√2)] = [(2+√2)²]/(4-2) = (2 + √2)²/2; so its square root is (2 + √2)/(√2). Similar considerations reduce the root of the second term to (2 - √2)/(√2). The difference is then (2√2)/(√2) = 2. Differences of two irrationals can equal a rational.

- Chris Maslanka

1 [(2+√2)/(2-√2)] can be written with a rational denominator by multiplying top and bottom by (2 + √2), giving [(2 + √2)(2+√2)]/[(2-√2)(2+√2)] = [(2+√2)²]/(4-2) = (2 + √2)²/2; so its square root is (2 + √2)/(√2). Similar considerations reduce the root of the second term to (2 - √2)/(√2). The difference is then (2√2)/(√2) = 2. Differences of two irrationals can equal a rational. Some educationalists argue that having to present answers with rational denominators is overly fussy; but the discipline of "rationalising the denominator" is useful prep for examples such as this one.