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3 Responses
You’re fine: for positive numbers, √a/√b = √(a/b), so there’s no need to rationalise first. Worked example: √50/√2 = √25 = 5 and 3√8/√2 = 3√4 = 6, so (√50 + 3√8)/√2 = 5 + 6 = 11.
You’re safe-no rule broken: rewrite √50 = 5√2 and √8 = 2√2, so (√50 + 3√8)/√2 = (5√2 + 3·2√2)/√2 = (5√2 + 6√2)/√2 = 11√2/√2 = 11.
Suspiciously neat, yes, but sometimes the surds line up like well-behaved ducks; rationalising isn’t needed here (I think!).
Your method is fine: you can split the fraction because both terms share the same denominator, and you can also use √a/√b = √(a/b) when a,b ≥ 0. So √50/√2 = √(50/2) = √25 = 5, and 3·√8/√2 = 3·√(8/2) = 3·√4 = 6, giving (√50 + 3√8)/√2 = 5 + 6 = 11. Rationalising first isn’t necessary here (you’d just do extra steps and still get 11). As a quick check with a simpler example: √18/√2 = √(18/2) = √9 = 3.