To solve the expression \((8/9) \times (-3/4) \times (1/9) \times (-4/5) \div (3/5)\), we'll proceed step by step, applying the rules of multiplication and division of fractions.
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Multiplication of Fractions: We start by multiplying the fractions together: \[ (8/9) \times (-3/4) \times (1/9) \times (-4/5) \]
First, we'll multiply the first two fractions: \[ (8/9) \times (-3/4) = \frac{8 \times -3}{9 \times 4} = \frac{-24}{36} = \frac{-2}{3} \] (simplifying \(-24/36\) gives us \(-2/3\)).
Next, multiply this result by the third fraction: \[ \frac{-2}{3} \times \frac{1}{9} = \frac{-2 \times 1}{3 \times 9} = \frac{-2}{27} \]
Finally, multiply by the fourth fraction: \[ \frac{-2}{27} \times \frac{-4}{5} = \frac{-2 \times -4}{27 \times 5} = \frac{8}{135} \]
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Dividing by a Fraction: Now we need to divide by \((3/5)\). Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{8}{135} \div \frac{3}{5} = \frac{8}{135} \times \frac{5}{3} \] Now multiply: \[ \frac{8 \times 5}{135 \times 3} = \frac{40}{405} \]
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Simplifying the Result: To simplify \(\frac{40}{405}\), we look for the greatest common divisor (GCD) of the numerator and denominator. The GCD of 40 and 405 is 5 (since \(40 = 5 \times 8\) and \(405 = 5 \times 81\)): \[ \frac{40 \div 5}{405 \div 5} = \frac{8}{81} \]
Thus, the final answer is: \[ \frac{8}{81} \]