L4 · Roots and Irrationals Clinic
“I can handle every trap in this unit mixed together: √ signs hiding integers, calculator displays pretending to end, roots of fractions and decimals, negative numbers on the line, and problems that want the exact value and the approximate value kept separate.”
Classify each as rational or irrational: (a) √0.25 (b) 0.01001000100001... (c) -7/3. Step 1: (a) Evaluate first. 0.5 x 0.5 = 0.25, so √0.25 = 0.5 exactly - a terminating decimal -> rational, radical costume and all. Step 2: (b) The blocks of 0s between the 1s grow by one each time - the decimal never terminates and never repeats a fixed block -> irrational. Step 3: (c) -7/3 is written as a quotient of integers, so it is rational by definition (its decimal, -2.333..., repeats). Say it plainly: the clinic rule from L1 still runs everything - classify the VALUE, using the decimal's behaviour or an exact evaluation, never the costume.
Why is √0.25 rational even though it wears a radical sign?
Classify √1.44. -> Rational: √1.44 = 1.2 exactly, because 1.2 x 1.2 = 1.44.
"√(49/81) = ___ / ___ = ___/___, so it is ___________." -> √49 / √81 = 7/9; rational
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.