If the decimal number \(0.15\overline{201}\) can be expressed in the form of \(\dfrac{a}{b}\) where a and b are co-primes, then find the value of \(\sqrt{b - a + 281}\).
120
Let d = 0.15\(\overline{201}\). Then 100000·d = 15201.201201… and 100·d = 15.201201…
Subtracting, 99900·d = 15186, so d = 15186/99900.
Divide numerator and denominator by 6: d = 2531/16650. Since 2531 is prime and does not divide 16650, this is in lowest terms, so a = 2531 and b = 16650.
b − a + 281 = 16650 − 2531 + 281 = 14400.
\(\sqrt{14400} = 120\).
Hence, the answer is 120.
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