Rooting Out Nine Tenths

In a mental-math round you are asked for 0.9 \sqrt{0.9} to three or four decimal places — no calculator, ten seconds on the clock. What is the value, and what is a fast route to it? (Beware the reflex answer 0.3 0.3 : that is 0.09 \sqrt{0.09} .)

Answer

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